Descriptive set theoryFrom Wikipedia, the free encyclopedia

Contents

1 Adequate pointclass 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Analytic set 22.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Projective hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Analytical hierarchy 43.1 The analytical hierarchy of formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 The analytical hierarchy of sets of natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 The analytical hierarchy on subsets of Cantor and Baire space . . . . . . . . . . . . . . . . . . . . 43.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Arithmetical hierarchy 74.1 The arithmetical hierarchy of formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 The arithmetical hierarchy of sets of natural numbers . . . . . . . . . . . . . . . . . . . . . . . . 84.3 Relativized arithmetical hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.4 Arithmetic reducibility and degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.5 The arithmetical hierarchy of subsets of Cantor and Baire space . . . . . . . . . . . . . . . . . . . 94.6 Extensions and variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.7 Meaning of the notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.9 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.10 Relation to Turing machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Arithmetical set 12

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5.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.4 Implicitly arithmetical sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Axiom of projective determinacy 146.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

7 Baire space (set theory) 157.1 Topology and trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.3 Relation to the real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8 Banach–Mazur game 178.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.2 A simple proof: winning strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

9 Borel equivalence relation 199.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.2 Kuratowski’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

10 Borel hierarchy 2010.1 Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2010.2 Boldface Borel hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

10.2.1 Borel sets of small rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2110.3 Lightface hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2110.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2210.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

11 Borel right process 23

12 Borel set 2512.1 Generating the Borel algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

12.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2612.2 Standard Borel spaces and Kuratowski theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 2612.3 Non-Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2612.4 Alternative non-equivalent definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2712.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2712.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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12.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2812.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

13 Cabal (set theory) 2913.1 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2913.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

14 Cantor set 3014.1 Construction and formula of the ternary set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3014.2 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3114.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

14.3.1 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3114.3.2 Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3214.3.3 Topological and analytical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3214.3.4 Measure and probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

14.4 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.4.1 Smith–Volterra–Cantor set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.4.2 Cantor dust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

14.5 Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3514.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

15 Cantor space 3615.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3615.2 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3615.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3715.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3715.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

16 Choquet game 3816.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

17 Cichoń's diagram 3917.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3917.2 Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4017.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4017.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

18 Coanalytic set 4118.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

19 Descriptive set theory 42

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19.1 Polish spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4219.1.1 Universality properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

19.2 Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4219.2.1 Borel hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4319.2.2 Regularity properties of Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

19.3 Analytic and coanalytic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4319.4 Projective sets and Wadge degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4319.5 Borel equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4419.6 Effective descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4419.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4419.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4419.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

20 Difference hierarchy 4520.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

21 Effective descriptive set theory 4621.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

21.1.1 Effective Polish space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4621.1.2 Arithmetical hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

21.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

22 Effective Polish space 4822.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4822.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

23 Fσ set 4923.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4923.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4923.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

24 Gregory number 5024.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5024.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

25 Gδ set 5125.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5125.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5125.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

25.3.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5225.4 Gδ space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5225.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5225.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5225.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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26 Homogeneous tree 5426.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

27 Homogeneously Suslin set 5527.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5527.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

28 Inductive set 5628.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

29 Infinity-Borel set 5729.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5729.2 Incorrect definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5729.3 Alternative characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5829.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

30 Interval (mathematics) 5930.1 Notations for intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

30.1.1 Including or excluding endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5930.1.2 Infinite endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6030.1.3 Integer intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

30.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6030.3 Classification of intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

30.3.1 Intervals of the extended real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6130.4 Properties of intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6130.5 Dyadic intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6230.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

30.6.1 Multi-dimensional intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6230.6.2 Complex intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

30.7 Topological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6230.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6330.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6330.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

31 Kleene–Brouwer order 6431.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6431.2 Tree interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6431.3 Recursion theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6531.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6531.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

32 Kuratowski–Ulam theorem 6632.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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33 L(R) 6733.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6733.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6733.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6733.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

34 Lightface analytic game 69

35 List of properties of sets of reals 7035.1 Definability properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7035.2 Regularity properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7035.3 Largeness and smallness properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

36 Lusin’s separation theorem 7236.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7236.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

37 Luzin space 7337.1 In real analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7337.2 Example of a Luzin set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7337.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

38 Martin measure 7538.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7538.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7538.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

39 Meagre set 7639.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

39.1.1 Relation to Borel hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7639.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7639.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7739.4 Banach–Mazur game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7739.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

39.5.1 Subsets of the reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7739.5.2 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

39.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7739.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7739.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

40 Negative-dimensional space 7940.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7940.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7940.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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40.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7940.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

41 Nested intervals 8141.1 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8241.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8241.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

42 Normal number 8342.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8342.2 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

42.2.1 Non-normal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8542.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

42.3 Connection to finite-state machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8642.4 Connection to equidistributed sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8742.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8742.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8842.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8842.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8942.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

43 Perfect set property 9043.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

44 Pointclass 9144.1 Basic framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9144.2 Boldface pointclasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9144.3 Lightface pointclasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9244.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

45 Polish group 9345.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9345.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9345.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

46 Polish space 9446.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9446.2 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9546.3 Polish metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9546.4 Generalizations of Polish spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

46.4.1 Lusin spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9546.4.2 Suslin spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9546.4.3 Radon spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9646.4.4 Polish groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

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46.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9646.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

47 Prewellordering 9747.1 Prewellordering property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

47.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9747.1.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

47.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9847.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

48 Projective hierarchy 9948.1 Relationship to the analytical hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9948.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

49 Property of Baire 10049.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10049.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10049.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

50 Scale (descriptive set theory) 10150.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10150.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10250.3 Scale property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10250.4 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10250.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10250.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

51 Schröder–Bernstein theorem for measurable spaces 10351.1 The theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

51.1.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10351.1.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

51.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10351.2.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10351.2.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

51.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

52 Smith–Volterra–Cantor set 10552.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10552.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10652.3 Other fat Cantor sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10652.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10652.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10652.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

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53 Stoneham number 10753.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

54 Suslin operation 10854.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10854.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

55 Tree (descriptive set theory) 10955.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

55.1.1 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10955.1.2 Branches and bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10955.1.3 Terminal nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

55.2 Relation to other types of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10955.3 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11055.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11055.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

56 Uniformization (set theory) 11156.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

57 Unit interval 11357.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

57.1.1 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11357.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11457.3 Fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11457.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11457.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

58 Universally Baire set 11558.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11558.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

59 Universally measurable set 11659.1 Finiteness condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11659.2 Example contrasting with Lebesgue measurability . . . . . . . . . . . . . . . . . . . . . . . . . . 11659.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

60 Vitali set 11860.1 Measurable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11860.2 Construction and proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11860.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11960.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11960.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

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61 Wadge hierarchy 12061.1 Wadge degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12061.2 Wadge and Lipschitz games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12061.3 Structure of the Wadge hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12161.4 Other notions of degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12161.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12161.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12161.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

62 Zero-dimensional space 12362.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12362.2 Properties of spaces with covering dimension zero . . . . . . . . . . . . . . . . . . . . . . . . . . 12362.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12362.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

63 Θ (set theory) 12563.1 Proof of existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12563.2 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 126

63.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12663.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12963.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Chapter 1

Adequate pointclass

In the mathematical field of descriptive set theory, a pointclass can be called adequate if it contains all recursivepointsets and is closed under recursive substitution, bounded universal and existential quantification and preimagesby recursive functions.[1][2]

1.1 References[1] Moschovakis, Y. N. (1987), Descriptive Set Theory, Studies in Logic and the Foundations of Mathematics, Elsevier, p. 158,

ISBN 9780080963198.

[2] Gabbay, Dov M.; Kanamori, Akihiro; Woods, John (2012), Sets and Extensions in the Twentieth Century, Handbook of theHistory of Logic 6, Elsevier, p. 465, ISBN 9780080930664.

1

Chapter 2

Analytic set

Not to be confused with Analytical set.For analytic sets in geometry, see analytic space.

In descriptive set theory, a subset of a Polish spaceX is an analytic set if it is a continuous image of a Polish space.These sets were first defined by Luzin (1917) and his student Souslin (1917).

2.1 Definition

There are several equivalent definitions of analytic set. The following conditions on a subspace A of a Polish spaceare equivalent:

• A is analytic.

• A is empty or a continuous image of the Baire space ωω.

• A is a Suslin space, in other words A is the image of a Polish space under a continuous mapping.

• A is the continuous image of a Borel set in a Polish space.

• A is a Suslin set, the image of the Suslin operation.

• There is a Polish space Y and a Borel set B ⊆ X × Y such that A is the projection of B ; that is,

A = x ∈ X|(∃y ∈ Y )⟨x, y⟩ ∈ B.

• A is the projection of a closed set in the cartesian product of X times the Baire space.

• A is the projection of a Gδ set in the cartesian product of X times the Cantor space.

An alternative characterization, in the specific, important, case that X is Baire space ωω, is that the analytic setsare precisely the projections of trees on ω × ω . Similarly, the analytic subsets of Cantor space 2ω are precisely theprojections of trees on 2× ω .

2.2 Properties

Analytic subsets of Polish spaces are closed under countable unions and intersections, continuous images, and inverseimages. The complement of an analytic set need not be analytic. Suslin proved that if the complement of an analyticset is analytic then the set is Borel. (Conversely any Borel set is analytic and Borel sets are closed under complements.)Luzin proved more generally that any two disjoint analytic sets are separated by a Borel set: in other words there isa Borel set containing one and disjoint from the other. This is sometimes called the “Luzin separability principle”(though it was implicit in the proof of Suslin’s theorem).

2

2.3. PROJECTIVE HIERARCHY 3

Analytic sets are always Lebesgue measurable (indeed, universally measurable) and have the property of Baire andthe perfect set property.

2.3 Projective hierarchy

Analytic sets are also calledΣ11 (see projective hierarchy). Note that the bold font in this symbol is not the Wikipedia

convention, but rather is used distinctively from its lightface counterpart Σ11 (see analytical hierarchy). The comple-

ments of analytic sets are called coanalytic sets, and the set of coanalytic sets is denoted by Π11 . The intersection

∆11 = Σ1

1 ∩Π11 is the set of Borel sets.

2.4 References• El'kin, A.G. (2001), “Analytic set”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• Efimov, B.A. (2001), “Luzin separability principles”, in Hazewinkel, Michiel, Encyclopedia of Mathematics,Springer, ISBN 978-1-55608-010-4

• Kechris, A. S. (1995), Classical Descriptive Set Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94374-9

• Luzin, N.N. (1917), “Sur la classification de M. Baire”, C.R. Acad. Sci. Paris Sér. I Math. 164: 91–94

• N.N. Lusin, “Leçons sur les ensembles analytiques et leurs applications”, Gauthier-Villars (1930)

• Moschovakis, Yiannis N. (1980), Descriptive Set Theory, North Holland, ISBN 0-444-70199-0

• Martin, Donald A.: Measurable cardinals and analytic games. “Fundamenta Mathematicae” 66 (1969/1970),p. 287-291.

• Souslin, M. (1917), “Sur une définition des ensembles mesurables B sans nombres transfinis”, Comptes RendusAcad. Sci. Paris 164: 88–91

Chapter 3

Analytical hierarchy

This article is about the classification of sets. For making complex decisions, see Analytic hierarchy process.

In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierar-chy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which canhave quantifiers over both the set of natural numbers, N , and over functions from N to N . The analytical hierarchyof sets classifies sets by the formulas that can be used to define them; it is the lightface version of the projectivehierarchy.

3.1 The analytical hierarchy of formulas

The notation Σ10 = Π1

0 = ∆10 indicates the class of formulas in the language of second-order arithmetic with no

set quantifiers. This language does not contain set parameters. The Greek letters here are lightface symbols, whichindicate this choice of language. Each corresponding boldface symbol denotes the corresponding class of formulasin the extended language with a parameter for each real; see projective hierarchy for details.A formula in the language of second-order arithmetic is defined to be Σ1

n+1 if it is logically equivalent to a formulaof the form ∃X1 · · · ∃Xkψ where ψ is Π1

n . A formula is defined to be Π1n+1 if it is logically equivalent to a formula

of the form ∀X1 · · · ∀Xkψ where ψ is Σ1n . This inductive definition defines the classes Σ1

n and Π1n for every natural

number n .Because every formula has a prenex normal form, every formula in the language of second-order arithmetic is Σ1

n

or Π1n for some n . Because meaningless quantifiers can be added to any formula, once a formula is given the

classification Σ1n or Π1

n for some n it will be given the classifications Σ1m and Π1

m for allm greater than n .

3.2 The analytical hierarchy of sets of natural numbers

A set of natural numbers is assigned the classification Σ1n if it is definable by a Σ1

n formula. The set is assignedthe classification Π1

n if it is definable by a Π1n formula. If the set is both Σ1

n and Π1n then it is given the additional

classification∆1n .

The ∆11 sets are called hyperarithmetical. An alternate classification of these sets by way of iterated computable

functionals is provided by hyperarithmetical theory.

3.3 The analytical hierarchy on subsets of Cantor and Baire space

The analytical hierarchy can be defined on any effective Polish space; the definition is particularly simple for Cantorand Baire space because they fit with the language of ordinary second-order arithmetic. Cantor space is the set ofall infinite sequences of 0s and 1s; Baire space is the set of all infinite sequences of natural numbers. These are bothPolish spaces.

4

3.4. EXTENSIONS 5

The ordinary axiomatization of second-order arithmetic uses a set-based language in which the set quantifiers cannaturally be viewed as quantifying over Cantor space. A subset of Cantor space is assigned the classification Σ1

n if itis definable by a Σ1

n formula. The set is assigned the classification Π1n if it is definable by a Π1

n formula. If the set isboth Σ1

n and Π1n then it is given the additional classification∆1

n .A subset of Baire space has a corresponding subset of Cantor space under the map that takes each function from ωto ω to the characteristic function of its graph. A subset of Baire space is given the classification Σ1

n , Π1n , or ∆1

n

if and only if the corresponding subset of Cantor space has the same classification. An equivalent definition of theanalytical hierarchy on Baire space is given by defining the analytical hierarchy of formulas using a functional versionof second-order arithmetic; then the analytical hierarchy on subsets of Cantor space can be defined from the hierarchyon Baire space. This alternate definition gives exactly the same classifications as the first definition.Because Cantor space is homeomorphic to any finite Cartesian power of itself, and Baire space is homeomorphic toany finite Cartesian power of itself, the analytical hierarchy applies equally well to finite Cartesian power of one ofthese spaces. A similar extension is possible for countable powers and to products of powers of Cantor space andpowers of Baire space.

3.4 Extensions

As is the case with the arithmetical hierarchy, a relativized version of the analytical hierarchy can be defined. Thelanguage is extended to add a constant set symbol A. A formula in the extended language is inductively defined to beΣ1,A

n orΠ1,An using the same inductive definition as above. Given a set Y , a set is defined to beΣ1,Y

n if it is definableby a Σ1,A

n formula in which the symbol A is interpreted as Y ; similar definitions for Π1,Yn and∆1,Y

n apply. The setsthat are Σ1,Y

n or Π1,Yn , for any parameter Y, are classified in the projective hierarchy.

3.5 Examples• The set of all natural numbers which are indices of computable ordinals is a Π1

1 set which is not Σ11 .

• The set of elements of Cantor space which are the characteristic functions of well orderings of ω is a Π11 set

which is not Σ11 . In fact, this set is not Σ1,Y

1 for any element Y of Baire space.

• If the axiom of constructibility holds then there is a subset of the product of the Baire space with itself which is∆1

2 and is the graph of a well ordering of Baire space. If the axiom holds then there is also a∆12 well ordering

of Cantor space.

3.6 Properties

For each n we have the following strict containments:

Π1n ⊂ Σ1

n+1

Π1n ⊂ Π1

n+1

Σ1n ⊂ Π1

n+1

Σ1n ⊂ Σ1

n+1

A set that is inΣ1n for some n is said to be analytical. Care is required to distinguish this usage from the term analytic

set which has a different meaning.

3.7 References• Rogers, H. (1967). Theory of recursive functions and effective computability. McGraw-Hill.• Kechris, A. (1995). Classical Descriptive Set Theory (Graduate Texts inMathematics 156 ed.). Springer. ISBN0-387-94374-9.

6 CHAPTER 3. ANALYTICAL HIERARCHY

3.8 External links• PlanetMath page

Chapter 4

Arithmetical hierarchy

In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy classi-fies certain sets based on the complexity of formulas that define them. Any set that receives a classification is calledarithmetical.The arithmetical hierarchy is important in recursion theory, effective descriptive set theory, and the study of formaltheories such as Peano arithmetic.The Tarski-Kuratowski algorithm provides an easy way to get an upper bound on the classifications assigned to aformula and the set it defines.The hyperarithmetical hierarchy and the analytical hierarchy extend the arithmetical hierarchy to classify additionalformulas and sets.

4.1 The arithmetical hierarchy of formulas

The arithmetical hierarchy assigns classifications to the formulas in the language of first-order arithmetic. The clas-sifications are denoted Σ0

n and Π0n for natural numbers n (including 0). The Greek letters here are lightface symbols,

which indicates that the formulas do not contain set parameters.If a formula ϕ is logically equivalent to a formula with only bounded quantifiers then ϕ is assigned the classificationsΣ0

0 and Π00 .

The classifications Σ0n and Π0

n are defined inductively for every natural number n using the following rules:

• If ϕ is logically equivalent to a formula of the form ∃n1∃n2 · · · ∃nkψ , where ψ is Π0n , then ϕ is assigned the

classification Σ0n+1 .

• If ϕ is logically equivalent to a formula of the form ∀n1∀n2 · · · ∀nkψ , where ψ is Σ0n , then ϕ is assigned the

classification Π0n+1 .

Also, a Σ0n formula is equivalent to a formula that begins with some existential quantifiers and alternates n− 1 times

between series of existential and universal quantifiers; while a Π0n formula is equivalent to a formula that begins with

some universal quantifiers and alternates similarly.Because every formula is equivalent to a formula in prenex normal form, every formula with no set quantifiers isassigned at least one classification. Because redundant quantifiers can be added to any formula, once a formula isassigned the classification Σ0

n or Π0n it will be assigned the classifications Σ0

m and Π0m for every m greater than n.

The most important classification assigned to a formula is thus the one with the least n, because this is enough todetermine all the other classifications.

7

8 CHAPTER 4. ARITHMETICAL HIERARCHY

4.2 The arithmetical hierarchy of sets of natural numbers

A set X of natural numbers is defined by formula φ in the language of Peano arithmetic (the first-order language withsymbols “0” for zero, “S” for the successor function, "+" for addition, "×" for multiplication, and "=" for equality), ifthe elements of X are exactly the numbers that satisfy φ. That is, for all natural numbers n,

n ∈ X ⇔ N |= ϕ(n),

where n is the numeral in the language of arithmetic corresponding to n . A set is definable in first order arithmeticif it is defined by some formula in the language of Peano arithmetic.Each set X of natural numbers that is definable in first order arithmetic is assigned classifications of the form Σ0

n ,Π0

n , and ∆0n , where n is a natural number, as follows. If X is definable by a Σ0

n formula then X is assigned theclassification Σ0

n . If X is definable by a Π0n formula then X is assigned the classification Π0

n . If X is both Σ0n and

Π0n then X is assigned the additional classification∆0

n .Note that it rarely makes sense to speak of ∆0

n formulas; the first quantifier of a formula is either existential oruniversal. So a∆0

n set is not defined by a∆0n formula; rather, there are both Σ0

n and Π0n formulas that define the set.

A parallel definition is used to define the arithmetical hierarchy on finite Cartesian powers of the natural numbers.Instead of formulas with one free variable, formulas with k free number variables are used to define the arithmeticalhierarchy on sets of k-tuples of natural numbers.

4.3 Relativized arithmetical hierarchies

Just as we can define what it means for a set X to be recursive relative to another set Y by allowing the computationdefining X to consult Y as an oracle we can extend this notion to the whole arithmetic hierarchy and define what itmeans for X to be Σ0

n , ∆0n or Π0

n in Y, denoted respectively Σ0,Yn ∆0,Y

n and Π0,Yn . To do so, fix a set of integers

Y and add a predicate for membership in Y to the language of Peano arithmetic. We then say that X is in Σ0,Yn if

it is defined by a Σ0n formula in this expanded language. In other words X is Σ0,Y

n if it is defined by a Σ0n formula

allowed to ask questions about membership in Y. Alternatively one can view the Σ0,Yn sets as those sets that can be

built starting with sets recursive in Y and alternately taking unions and intersections of these sets up to n times.For example let Y be a set of integers. Let X be the set of numbers divisible by an element of Y. Then X is definedby the formula ϕ(n) = ∃m∃t(Y (m) ∧m × t = n) so X is in Σ0,Y

1 (actually it is in ∆0,Y0 as well since we could

bound both quantifiers by n).

4.4 Arithmetic reducibility and degrees

Arithmetical reducibility is an intermediate notion between Turing reducibility and hyperarithmetic reducibility.A set is arithmetical (also arithmetic and arithmetically definable) if it is defined by some formula in the languageof Peano arithmetic. Equivalently X is arithmetical if X is Σ0

n or Π0n for some integer n. A set X is arithmetical

in a set Y, denoted X ≤A Y , if X is definable a some formula in the language of Peano arithmetic extended by apredicate for membership in Y. Equivalently, X is arithmetical in Y if X is in Σ0,Y

n or Π0,Yn for some integer n. A

synonym for X ≤A Y is: X is arithmetically reducible to Y.The relationX ≤A Y is reflexive and transitive, and thus the relation ≡A defined by the rule

X ≡A Y ⇔ X ≤A Y ∧ Y ≤A X

is an equivalence relation. The equivalence classes of this relation are called the arithmetic degrees; they are partiallyordered under ≤A .

4.5. THE ARITHMETICAL HIERARCHY OF SUBSETS OF CANTOR AND BAIRE SPACE 9

4.5 The arithmetical hierarchy of subsets of Cantor and Baire space

The Cantor space, denoted 2ω , is the set of all infinite sequences of 0s and 1s; the Baire space, denoted ωω or N ,is the set of all infinite sequences of natural numbers. Note that elements of the Cantor space can be identified withsets of integers and elements of the Baire space with functions from integers to integers.The ordinary axiomatization of second-order arithmetic uses a set-based language in which the set quantifiers cannaturally be viewed as quantifying over Cantor space. A subset of Cantor space is assigned the classificationΣ0

n if it isdefinable by aΣ0

n formula. The set is assigned the classificationΠ0n if it is definable by aΠ0

n formula. If the set is bothΣ0

n and Π0n then it is given the additional classification∆0

n . For example let O ⊂ 2ω be the set of all infinite binarystrings which aren't all 0 (or equivalently the set of all non-empty sets of integers). AsO = X ∈ 2ω|∃n(X(n) = 1)we see that O is defined by a Σ0

1 formula and hence is a Σ01 set.

Note that while both the elements of the Cantor space (regarded as sets of integers) and subsets of the Cantor spaceare classified in arithmetic hierarchies, these are not the same hierarchy. In fact the relationship between the twohierarchies is interesting and non-trivial. For instance the Π0

n elements of the Cantor space are not (in general)the same as the elements X of the Cantor space so that X is a Π0

n subset of the Cantor space. However, manyinteresting results relate the two hierarchies.There are two ways that a subset of Baire space can be classified in the arithmetical hierarchy.

• A subset of Baire space has a corresponding subset of Cantor space under the map that takes each functionfrom ω to ω to the characteristic function of its graph. A subset of Baire space is given the classification Σ1

n ,Π1

n , or∆1n if and only if the corresponding subset of Cantor space has the same classification.

• An equivalent definition of the analytical hierarchy on Baire space is given by defining the analytical hierarchyof formulas using a functional version of second-order arithmetic; then the analytical hierarchy on subsets ofCantor space can be defined from the hierarchy on Baire space. This alternate definition gives exactly the sameclassifications as the first definition.

A parallel definition is used to define the arithmetical hierarchy on finite Cartesian powers of Baire space or Cantorspace, using formulas with several free variables. The arithmetical hierarchy can be defined on any effective Polishspace; the definition is particularly simple for Cantor space and Baire space because they fit with the language ofordinary second-order arithmetic.Note that we can also define the arithmetic hierarchy of subsets of the Cantor and Baire spaces relative to some setof integers. In fact boldface 0

n is just the union of Σ0,Yn for all sets of integers Y. Note that the boldface hierarchy

is just the standard hierarchy of Borel sets.

4.6 Extensions and variations

It is possible to define the arithmetical hierarchy of formulas using a language extended with a function symbol foreach primitive recursive function. This variation slightly changes the classification of some sets.Amore semantic variation of the hierarchy can be defined on all finitary relations on the natural numbers; the followingdefinition is used. Every computable relation is defined to be Σ0

0 and Π00 . The classifications Σ0

n and Π0n are defined

inductively with the following rules.

• If the relationR(n1, . . . , nl,m1, . . . ,mk) isΣ0n then the relationS(n1, . . . , nl) = ∀m1 · · · ∀mkR(n1, . . . , nl,m1, . . . ,mk)

is defined to be Π0n+1

• If the relationR(n1, . . . , nl,m1, . . . ,mk) isΠ0n then the relationS(n1, . . . , nl) = ∃m1 · · · ∃mkR(n1, . . . , nl,m1, . . . ,mk)

is defined to be Σ0n+1

This variation slightly changes the classification of some sets. It can be extended to cover finitary relations on thenatural numbers, Baire space, and Cantor space.

10 CHAPTER 4. ARITHMETICAL HIERARCHY

4.7 Meaning of the notation

The following meanings can be attached to the notation for the arithmetical hierarchy on formulas.The subscript n in the symbols Σ0

n and Π0n indicates the number of alternations of blocks of universal and existential

number quantifiers that are used in a formula. Moreover, the outermost block is existential in Σ0n formulas and

universal in Π0n formulas.

The superscript 0 in the symbols Σ0n , Π0

n , and ∆0n indicates the type of the objects being quantified over. Type

0 objects are natural numbers, and objects of type i + 1 are functions that map the set of objects of type i to thenatural numbers. Quantification over higher type objects, such as functions from natural numbers to natural numbers,is described by a superscript greater than 0, as in the analytical hierarchy. The superscript 0 indicates quantifiers overnumbers, the superscript 1 would indicate quantification over functions from numbers to numbers (type 1 objects),the superscript 2 would correspond to quantification over functions that take a type 1 object and return a number, andso on.

4.8 Examples

• The Σ01 sets of numbers are those definable by a formula of the form ∃n1 · · · ∃nkψ(n1, . . . , nk,m) where ψ

has only bounded quantifiers. These are exactly the recursively enumerable sets.

• The set of natural numbers that are indices for Turing machines that compute total functions isΠ02 . Intuitively,

an index e falls into this set if and only if for every m “there is an s such that the Turing machine with indexe halts on input m after s steps”. A complete proof would show that the property displayed in quotes in theprevious sentence is definable in the language of Peano arithmetic by a Σ0

1 formula.

• Every Σ01 subset of Baire space or Cantor space is an open set in the usual topology on the space. Moreover,

for any such set there is a computable enumeration of Gödel numbers of basic open sets whose union is theoriginal set. For this reason, Σ0

1 sets are sometimes called effectively open. Similarly, every Π01 set is closed

and the Π01 sets are sometimes called effectively closed.

• Every arithmetical subset of Cantor space or Baire space is a Borel set. The lightface Borel hierarchy extendsthe arithmetical hierarchy to include additional Borel sets. For example, every Π0

2 subset of Cantor or Bairespace is a Gδ set (that is, a set which equals the intersection of countably many open sets). Moreover, eachof these open sets is Σ0

1 and the list of Gödel numbers of these open sets has a computable enumeration.If ϕ(X,n,m) is a Σ0

0 formula with a free set variable X and free number variables n,m then the Π02 set

X | ∀n∃mϕ(X,n,m) is the intersection of the Σ01 sets of the form X | ∃mϕ(X,n,m) as n ranges over

the set of natural numbers.

4.9 Properties

The following properties hold for the arithmetical hierarchy of sets of natural numbers and the arithmetical hierarchyof subsets of Cantor or Baire space.

• The collections Π0n and Σ0

n are closed under finite unions and finite intersections of their respective elements.

• A set isΣ0n if and only if its complement isΠ0

n . A set is∆0n if and only if the set is bothΣ0

n andΠ0n , in which

case its complement will also be ∆0n .

• The inclusions∆0n ⊊ Π0

n and∆0n ⊊ Σ0

n hold for n ≥ 1 .

• The inclusions Π0n ⊊ Π0

n+1 and Σ0n ⊊ Σ0

n+1 hold for all n and the inclusion Σ0n ∪ Π0

n ⊊ ∆0n+1 holds for

n ≥ 1 . Thus the hierarchy does not collapse.

4.10. RELATION TO TURING MACHINES 11

4.10 Relation to Turing machines

The Turing computable sets of natural numbers are exactly the sets at level ∆01 of the arithmetical hierarchy. The

recursively enumerable sets are exactly the sets at level Σ01 .

No oracle machine is capable of solving its own halting problem (a variation of Turing’s proof applies). The haltingproblem for a∆0,Y

n oracle in fact sits in Σ0,Yn+1 .

Post’s theorem establishes a close connection between the arithmetical hierarchy of sets of natural numbers and theTuring degrees. In particular, it establishes the following facts for all n ≥ 1:

• The set ∅(n) (the nth Turing jump of the empty set) is many-one complete in Σ0n .

• The set N \ ∅(n) is many-one complete in Π0n .

• The set ∅(n−1) is Turing complete in∆0n .

The polynomial hierarchy is a “feasible resource-bounded” version of the arithmetical hierarchy in which polyno-mial length bounds are placed on the numbers involved (or, equivalently, polynomial time bounds are placed on theTuring machines involved). It gives a finer classification of some sets of natural numbers that are at level ∆0

1 of thearithmetical hierarchy.

4.11 See also• Interpretability logic

• Hierarchy (mathematics)

• Polynomial hierarchy

4.12 References• Japaridze, Giorgie (1994), “The logic of arithmetical hierarchy”, Annals of Pure and Applied Logic 66 (2):89–112, doi:10.1016/0168-0072(94)90063-9, Zbl 0804.03045.

• Moschovakis, Yiannis N. (1980), Descriptive Set Theory, Studies in Logic and the Foundations of Mathematics100, North Holland, ISBN 0-444-70199-0, Zbl 0433.03025.

• Nies, André (2009), Computability and randomness, Oxford Logic Guides 51, Oxford: Oxford UniversityPress, ISBN 978-0-19-923076-1, Zbl 1169.03034.

• Rogers, H., jr (1967), Theory of recursive functions and effective computability, Maidenhead: McGraw-Hill,Zbl 0183.01401.

Chapter 5

Arithmetical set

In mathematical logic, an arithmetical set (or arithmetic set) is a set of natural numbers that can be defined by aformula of first-order Peano arithmetic. The arithmetical sets are classified by the arithmetical hierarchy.The definition can be extended to an arbitrary countable set A (e.g. the set of n-tuples of integers, the set of rationalnumbers, the set of formulas in some formal language, etc.) by using Gödel numbers to represent elements of the setand declaring a subset of A to be arithmetical if the set of corresponding Gödel numbers is arithmetical.A function f :⊆ Nk → N is called arithmetically definable if the graph of f is an arithmetical set.A real number is called arithmetical if the set of all smaller rational numbers is arithmetical. A complex number iscalled arithmetical if its real and imaginary parts are both arithmetical.

5.1 Formal definition

A set X of natural numbers is arithmetical or arithmetically definable if there is a formula φ(n) in the lan-guage of Peano arithmetic such that each number n is in X if and only if φ(n) holds in the standard model ofarithmetic. Similarly, a k-ary relation R(n1, . . . , nk) is arithmetical if there is a formula ψ(n1, . . . , nk) such thatR(n1, . . . , nk) ⇔ ψ(n1, . . . , nk) holds for all k-tuples (n1, . . . , nk) of natural numbers.A finitary function on the natural numbers is called arithmetical if its graph is an arithmetical binary relation.A setA is said to be arithmetical in a set B ifA is definable by an arithmetical formula which has B as a set parameter.

5.2 Examples• The set of all prime numbers is arithmetical.

• Every recursively enumerable set is arithmetical.

• Every computable function is arithmetically definable.

• The set encoding the Halting problem is arithmetical.

• Chaitin’s constant Ω is an arithmetical real number.

• Tarski’s indefinability theorem shows that the set of true formulas of first order arithmetic is not arithmeticallydefinable.

5.3 Properties• The complement of an arithmetical set is an arithmetical set.

• The Turing jump of an arithmetical set is an arithmetical set.

12

5.4. IMPLICITLY ARITHMETICAL SETS 13

• The collection of arithmetical sets is countable, but there is no arithmetically definable sequence that enumeratesall arithmetical sets.

• The set of real arithmetical numbers is countable, dense and order-isomorphic to the set of rational numbers.

5.4 Implicitly arithmetical sets

Each arithmetical set has an arithmetical formula which tells whether particular numbers are in the set. An alternativenotion of definability allows for a formula that does not tell whether particular numbers are in the set but tells whetherthe set itself satisfies some arithmetical property.A set Y of natural numbers is implicitly arithmetical or implicitly arithmetically definable if it is definable withan arithmetical formula that is able to use Y as a parameter. That is, if there is a formula θ(Z) in the language ofPeano arithmetic with no free number variables and a new set parameter Z and set membership relation ∈ such thatY is the unique set Z such that θ(Z) holds.Every arithmetical set is implicitly arithmetical; if X is arithmetically defined by φ(n) then it is implicitly defined bythe formula

∀n[n ∈ Z ⇔ ϕ(n)]

Not every implicitly arithmetical set is arithmetical, however. In particular, the truth set of first order arithmetic isimplicitly arithmetical but not arithmetical.

5.5 See also• Arithmetical hierarchy

• Computable set

• Computable number

5.6 Further reading• Rogers, H. (1967). Theory of recursive functions and effective computability. McGraw-Hill. OCLC 527706

Chapter 6

Axiom of projective determinacy

In mathematical logic, projective determinacy is the special case of the axiom of determinacy applying only toprojective sets.The axiom of projective determinacy, abbreviated PD, states that for any two-player game of perfect informationof length ω in which the players play natural numbers, if the victory set (for either player, since the projective setsare closed under complementation) is projective, then one player or the other has a winning strategy.The axiom is not a theorem of ZFC (assuming ZFC is consistent), but unlike the full axiom of determinacy (AD),which contradicts the axiom of choice, it is not known to be inconsistent with ZFC. PD follows from certain largecardinal axioms, such as the existence of infinitely many Woodin cardinals.PD implies that all projective sets are Lebesgue measurable (in fact, universally measurable) and have the perfectset property and the property of Baire. It also implies that every projective binary relation may be uniformized by aprojective set.

6.1 References• Martin, Donald A. and John R. Steel (Jan 1989). “A Proof of Projective Determinacy”. Journal of the Amer-

ican Mathematical Society (American Mathematical Society) 2 (1): 71–125. doi:10.2307/1990913. JSTOR1990913.

• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

14

Chapter 7

Baire space (set theory)

For the concept in topology, see Baire space.

In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology. Thisspace is commonly used in descriptive set theory, to the extent that its elements are often called “reals.” It is denotedB, NN, ωω, ωω, or N .The Baire space is defined to be the Cartesian product of countably infinitely many copies of the set of naturalnumbers, and is given the product topology (where each copy of the set of natural numbers is given the discretetopology). The Baire space is often represented using the tree of finite sequences of natural numbers.The Baire space can be contrasted with Cantor space, the set of infinite sequences of binary digits.

7.1 Topology and trees

The product topology used to define the Baire space can be described more concretely in terms of trees. The basicopen sets of the product topology are cylinder sets, here characterized as:

If any finite set of natural number coordinates ci : i < n is selected, and for each ci a particular naturalnumber value vi is selected, then the set of all infinite sequences of natural numbers that have value viat position ci for all i < n is a basic open set. Every open set is a union of a collection of these.

By moving to a different basis for the same topology, an alternate characterization of open sets can be obtained:

If a sequence of natural numbers wi : i < n is selected, then the set of all infinite sequences of naturalnumbers that have value wi at position i for all i < n is a basic open set. Every open set is a union of acollection of these.

Thus a basic open set in the Baire space specifies a finite initial segment τ of an infinite sequence of natural numbers,and all the infinite sequences extending τ form a basic open set. This leads to a representation of the Baire space asthe set of all paths through the full tree ω<ω of finite sequences of natural numbers ordered by extension. An openset is determined by some (possibly infinite) union of nodes of the tree; a point in Baire space is in the open set ifand only if its path goes through one of these nodes.The representation of the Baire space as paths through a tree also gives a characterization of closed sets. For anyclosed subset C of Baire space there is a subtree T of ω<ω such that any point x is in C if and only if x is a paththrough T. Conversely, the set of paths through any subtree of ω<ω is a closed set.

7.2 Properties

The Baire space has the following properties:

15

16 CHAPTER 7. BAIRE SPACE (SET THEORY)

1. It is a perfect Polish space, which means it is a completely metrizable second countable space with no isolatedpoints. As such, it has the same cardinality as the real line and is a Baire space in the topological sense of theterm.

2. It is zero-dimensional and totally disconnected.

3. It is not locally compact.

4. It is universal for Polish spaces in the sense that it can be mapped continuously onto any non-empty Polishspace. Moreover, any Polish space has a dense Gδ subspace homeomorphic to a Gδ subspace of the Bairespace.

5. The Baire space is homeomorphic to the product of any finite or countable number of copies of itself.

7.3 Relation to the real line

The Baire space is homeomorphic to the set of irrational numbers when they are given the subspace topology inheritedfrom the real line. A homeomorphism between Baire space and the irrationals can be constructed using continuedfractions.From the point of view of descriptive set theory, the fact that the real line is connected causes technical difficulties.For this reason, it is more common to study Baire space. Because every Polish space is the continuous image of Bairespace, it is often possible to prove results about arbitrary Polish spaces by showing that these properties hold for Bairespace and by showing that they are preserved by continuous functions.B is also of independent, but minor, interest in real analysis, where it is considered as a uniform space. The uniformstructures ofB and Ir (the irrationals) are different, however: B is complete in its usual metric while Ir is not (althoughthese spaces are homeomorphic).

7.4 References• Kechris, Alexander S. (1994). Classical Descriptive Set Theory. Springer-Verlag. ISBN 0-387-94374-9.

• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

Chapter 8

Banach–Mazur game

In general topology, set theory and game theory, aBanach–Mazur game is a topological game played by two players,trying to pin down elements in a set (space). The concept of a Banach–Mazur game is closely related to the concept ofBaire spaces. This game was the first infinite positional game of perfect information to be studied. It was introducedby Mazur as problem 43 in the Scottish book, and Mazur’s questions about it were answered by Banach.

8.1 Definition and properties

In what follows we will make use of the formalism defined in Topological game. A general Banach–Mazur game isdefined as follows: we have a topological space Y , a fixed subset X ⊂ Y , and a family W of subsets of Y thatsatisfy the following properties.

• Each member ofW has non-empty interior.

• Each non-empty open subset of Y contains a member ofW .

We will call this gameMB(X,Y,W ) . Two players, P1 and P2 , choose alternatively elementsW0 ,W1 , · · · ofWsuch thatW0 ⊃W1 ⊃ · · · . The player P1 wins if and only ifX ∩ (∩n<ωWn) = ∅ .The following properties hold.

• P2 ↑ MB(X,Y,W ) if and only if X is of the first category in Y (a set is of the first category or meagre if itis the countable union of nowhere-dense sets).

• Assuming that Y is a complete metric space, P1 ↑ MS(X,Y,W ) if and only if X is comeager in somenonempty open subset of Y .

• If X has the Baire property in Y , thenMB(X,Y,W ) is determined.

• Any winning strategy of P2 can be reduced to a stationary winning strategy.

• The siftable and strongly-siftable spaces introduced by Choquet can be defined in terms of stationary strategiesin suitable modifications of the game. Let BM(X) denote a modification ofMB(X,Y,W ) where X = Y ,W is the family of all nonempty open sets inX , andP2 wins a play (W0,W1, · · · ) if and only if∩n<ωWn = ∅. Then X is siftable if and only if P2 has a stationary winning strategy in BM(X) .

• A Markov winning strategy for P2 in BM(X) can be reduced to a stationary winning strategy. Furthermore,if P2 has a winning strategy in BM(X) , then she has a winning strategy depending only on two precedingmoves. It is still an unsettled question whether a winning strategy for P2 can be reduced to a winning strategythat depends only on the last two moves of P1 .

• X is called weakly α -favorable if P2 has a winning strategy in BM(X) . Then, X is a Baire space if andonly if P1 has no winning strategy inBM(X) . It follows that each weakly α -favorable space is a Baire space.

17

18 CHAPTER 8. BANACH–MAZUR GAME

Many other modifications and specializations of the basic game have been proposed: for a thorough account of these,refer to [1987]. The most common special case, calledMB(X, J) , consists in letting Y = J , i.e. the unit interval[0, 1] , and in letting W consist of all closed intervals [a, b] contained in [0, 1] . The players choose alternativelysubintervals J0, J1, · · · of J such that J0 ⊃ J1 ⊃ · · · , and P1 wins if and only if X ∩ (∩n<ωJn) = ∅ . P2 wins ifand only if X ∩ (∩n<ωJn) = ∅ .

8.2 A simple proof: winning strategies

It is natural to ask for what setsX does P2 have a winning strategy. Clearly, ifX is empty, P2 has a winning strategy,therefore the question can be informally rephrased as how “small” (respectively, “big”) does X (respectively, thecomplement ofX in Y ) have to be to ensure that P2 has a winning strategy. To give a flavor of how the proofs usedto derive the properties in the previous section work, let us show the following fact.Fact: P2 has a winning strategy if X is countable, Y is T1, and Y has no isolated points.

Proof: Let the elements ofX be x1, x2, · · · . Suppose thatW1 has been chosen byP1 , and letU1 be the (non-empty)interior of W1 . Then U1 \ x1 is a non-empty open set in Y , so P2 can choose a member W2 of W containedin this set. Then P1 chooses a subsetW3 ofW2 and, in a similar fashion, P2 can choose a memberW4 ⊂ W3 thatexcludes x2 . Continuing in this way, each point xn will be excluded by the setW2n , so that the intersection of alltheWn will have empty intersection with X . Q.E.DThe assumptions on Y are key to the proof: for instance, if Y = a, b, c is equipped with the discrete topology andW consists of all non-empty subsets of Y , then P2 has no winning strategy if X = a (as a matter of fact, heropponent has a winning strategy). Similar effects happen if Y is equipped with indiscrete topology andW = Y .A stronger result relatesX to first-order sets.Fact: Let Y be a topological space, letW be a family of subsets of Y satisfying the two properties above, and letXbe any subset of Y . P2 has a winning strategy if and only if X is meagre.This does not imply that P1 has a winning strategy ifX is not meagre. In fact, P1 has a winning strategy if and onlyif there is some Wi ∈ W such that X ∩Wi is a comeagre subset of Wi . It may be the case that neither playerhas a winning strategy: when Y is [0, 1] andW consists of the closed intervals [a, b] , the game is determined if thetarget set has the property of Baire, i.e. if it differs from an open set by a meagre set (but the converse is not true).Assuming the axiom of choice, there are subsets of [0, 1] for which the Banach–Mazur game is not determined.

8.3 References• [1957] Oxtoby, J.C. The Banach–Mazur game and Banach category theorem, Contribution to the Theory ofGames, Volume III, Annals of Mathematical Studies 39 (1957), Princeton, 159–163

• [1987] Telgársky, R. J. Topological Games: On the 50th Anniversary of the Banach–Mazur Game, RockyMountain J. Math. 17 (1987), pp. 227–276. (3.19 MB)

• [2003] Julian P. Revalski The Banach–Mazur game: History and recent developments, Seminar notes, Pointe-a-Pitre, Guadeloupe, France, 2003–2004

Chapter 9

Borel equivalence relation

In mathematics, a Borel equivalence relation on a Polish space X is an equivalence relation on X that is a Borelsubset of X × X (in the product topology).

9.1 Formal definition

Given Borel equivalence relations E and F on Polish spaces X and Y respectively, one says that E is Borel reducibleto F, in symbols E ≤B F, if and only if there is a Borel function

Θ : X → Y

such that for all x,x' ∈ X, one has

xEx' ⇔ Θ(x)FΘ(x' ).

Conceptually, if E is Borel reducible to F, then E is “not more complicated” than F, and the quotient space X/E hasa lesser or equal “Borel cardinality” than Y/F, where “Borel cardinality” is like cardinality except for a definabilityrestriction on the witnessing mapping.

9.2 Kuratowski’s theorem

A measure space X is called a standard Borel space if it is Borel-isomorphic to a Borel subset of a Polish space.Kuratowski’s theorem then states that two standard Borel spaces X and Y are Borel-isomorphic iff |X| = |Y |.

9.3 References• Harrington, L. A., A. S. Kechris, A. Louveau (Oct 1990). “A Glimm-Effros Dichotomy for Borel equivalencerelations”. Journal of the American Mathematical Society (Journal of the American Mathematical Society, Vol.3, No. 4) 3 (2): 903–928. doi:10.2307/1990906. JSTOR 1990906.

• Kechris, Alexander S. (1994). Classical Descriptive Set Theory. Springer-Verlag. ISBN 0-387-94374-9.

• Silver, Jack H. (1980). “Counting the number of equivalence classes of Borel and coanalytic equivalencerelations”. Annals of Mathematical Logic 18 (1): 1–28. doi:10.1016/0003-4843(80)90002-9.

• Kanovei, Vladimir; Borel equivalence relations. Structure and classification. University Lecture Series, 44.American Mathematical Society, Providence, RI, 2008. x+240 pp. ISBN 978-0-8218-4453-3

19

Chapter 10

Borel hierarchy

In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets ofa Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinalnumber called the rank of the Borel set. The Borel hierarchy is of particular interest in descriptive set theory.One common use of the Borel hierarchy is to prove facts about the Borel sets using transfinite induction on rank.Properties of sets of small finite ranks are important in measure theory and analysis.

10.1 Borel sets

Main article: Borel set

The Borel algebra in an arbitrary topological space is the smallest collection of subsets of the space that contains theopen sets and is closed under countable unions and complementation. It can be shown that the Borel algebra is closedunder countable intersections as well.A short proof that the Borel algebra is well defined proceeds by showing that the entire powerset of the space is closedunder complements and countable unions, and thus the Borel algebra is the intersection of all families of subsets ofthe space that have these closure properties. This proof does not give a simple procedure for determining whether aset is Borel. A motivation for the Borel hierarchy is to provide a more explicit characterization of the Borel sets.

10.2 Boldface Borel hierarchy

The Borel hierarchy or boldface Borel hierarchy on a space X consists of classes 0α , 0

α , and 0α for every

countable ordinalα greater than zero. Each of these classes consists of subsets ofX. The classes are defined inductivelyfrom the following rules:

• A set is in 01 if and only if it is open.

• A set is in 0α if and only if its complement is in 0

α .

• A set A is in 0α for α > 1 if and only if there is a sequence of sets A1, A2, . . . such that each Ai is in 0

αifor

some αi < α and A =∪Ai .

• A set is in 0α if and only if it is both in 0

α and in 0α .

The motivation for the hierarchy is to follow the way in which a Borel set could be constructed from open sets usingcomplementation and countable unions. A Borel set is said to have finite rank if it is in 0

α for some finite ordinalα; otherwise it has infinite rank.The hierarchy can be shown to have the following properties:

•∪

α<ω1

0α =

∪α<ω1

0α =

∪α<ω1

0α . Moreover, a set is in this union if and only if it is Borel.

20

10.3. LIGHTFACE HIERARCHY 21

• For every α, 0α ∪ 0

α ⊆ 0α+1 . Thus, once a set is in 0

α or 0α , that set will be in all classes in the hierarchy

corresponding to ordinals greater than α

• IfX is an uncountable Polish space, it can be shown that 0α is not contained in 0

α for any α < ω1 , and thusthe hierarchy does not collapse.

10.2.1 Borel sets of small rank

The classes of small rank are known by alternate names in classical descriptive set theory.

• The 01 sets are the open sets. The 0

1 sets are the closed sets.

• The 02 sets are countable unions of closed sets, and are called Fσ sets. The 0

2 sets are the dual class, and canbe written as a countable intersection of open sets. These sets are called Gδ sets.

10.3 Lightface hierarchy

The lightface Borel hierarchy is an effective version of the boldface Borel hierarchy. It is important in effectivedescriptive set theory and recursion theory. The lightface Borel hierarchy extends the arithmetical hierarchy of subsetsof an effective Polish space. It is closely related to the hyperarithmetical hierarchy.The lightface Borel hierarchy can be defined on any effective Polish space. It consists of classes Σ0

α , Π0α and ∆0

α

for each nonzero countable ordinal α less than the Church-Kleene ordinal ωCK1 . Each class consists of subsets of the

space. The classes, and codes for elements of the classes, are inductively defined as follows:

• A set isΣ01 if and only if it is effectively open, that is, an open set which is the union of a computably enumerable

sequence of basic open sets. A code for such a set is a pair (0,e), where e is the index of a program enumeratingthe sequence of basic open sets.

• A set is Π0α if and only if its complement is Σ0

α . A code for one of these sets is a pair (1,c) where c is a codefor the complementary set.

• A set is Σ0α if there is a computably enumerable sequence of codes for a sequenceA1, A2, . . . of sets such that

each Ai is Π0αi

for some αi < α and A =∪Ai . A code for a Σ0

α set is a pair (2,e), where e is an index of aprogram enumerating the codes of the sequence Ai .

A code for a lightface Borel set gives complete information about how to recover the set from sets of smaller rank.This contrasts with the boldface hierarchy, where no such effectivity is required. Each lightface Borel set has infinitelymany distinct codes. Other coding systems are possible; the crucial idea is that a code must effectively distinguishbetween effectively open sets, complements of sets represented by previous codes, and computable enumerations ofsequences of codes.It can be shown that for each α < ωCK

1 there are sets in Σ0α \Π0

α , and thus the hierarchy does not collapse. No newsets would be added at stage ωCK

1 , however.A famous theorem due to Spector and Kleene states that a set is in the lightface Borel hierarchy if and only if it is atlevel∆1

1 of the analytical hierarchy. These sets are also called hyperarithmetic.The code for a lightface Borel set A can be used to inductively define a tree whose nodes are labeled by codes. Theroot of the tree is labeled by the code for A. If a node is labeled by a code of the form (1,c) then it has a child nodewhose code is c. If a node is labeled by a code of the form (2,e) then it has one child for each code enumerated by theprogram with index e. If a node is labeled with a code of the form (0,e) then it has no children. This tree describeshow A is built from sets of smaller rank. The ordinals used in the construction of A ensure that this tree has no infinitepath, because any infinite path through the tree would have to include infinitely many codes starting with 2, and thuswould give an infinite decreasing sequence of ordinals. Conversely, if an arbitrary subtree of ω<ω has its nodeslabeled by codes in a consistent way, and the tree has no infinite paths, then the code at the root of the tree is a codefor a lightface Borel set. The rank of this set is bounded by the order type of the tree in the Kleene–Brouwer order.Because the tree is arithmetically definable, this rank must be less than ωCK

1 . This is the origin of the Church-Kleeneordinal in the definition of the lightface hierarchy.

22 CHAPTER 10. BOREL HIERARCHY

10.4 References• Kechris, Alexander. Classical Descriptive Set Theory. Graduate Texts in Mathematics v. 156, Springer-Verlag,1995. ISBN 3-540-94374-9.

• Jech, Thomas. Set Theory, 3rd edition. Springer, 2003. ISBN 3-540-44085-2.

10.5 See also• Wadge hierarchy

• Veblen hierarchy

Chapter 11

Borel right process

In the mathematical theory of probability, a Borel right process, named after Émile Borel, is a particular kind ofcontinuous-time random process.Let E be a locally compact, separable, metric space. We denote by E the Borel subsets of E . Let Ω be the spaceof right continuous maps from [0,∞) to E that have left limits in E , and for each t ∈ [0,∞) , denote by Xt thecoordinate map at t ; for each ω ∈ Ω , Xt(ω) ∈ E is the value of ω at t . We denote the universal completion of Eby E∗ . For each t ∈ [0,∞) , let

Ft = σX−1

s (B) : s ∈ [0, t], B ∈ E,

F∗t = σ

X−1

s (B) : s ∈ [0, t], B ∈ E∗ ,and then, let

F∞ = σX−1

s (B) : s ∈ [0,∞), B ∈ E,

F∗∞ = σ

X−1

s (B) : s ∈ [0,∞), B ∈ E∗ .For each Borel measurable function f on E , define, for each x ∈ E ,

Uαf(x) = Ex

[∫ ∞

0

e−αtf(Xt) dt

].

Since Ptf(x) = Ex [f(Xt)] and the mapping given by t → Xt is right continuous, we see that for any uniformlycontinuous function f , we have the mapping given by t→ Ptf(x) is right continuous.Therefore, together with the monotone class theorem, for any universally measurable function f , the mapping givenby (t, x) → Ptf(x) , is jointly measurable, that is, B([0,∞))⊗E∗ measurable, and subsequently, the mapping is also(B([0,∞))⊗ E∗)

λ⊗µ -measurable for all finite measures λ on B([0,∞)) and µ on E∗ . Here, (B([0,∞))⊗ E∗)λ⊗µ

is the completion of B([0,∞))⊗ E∗ with respect to the product measure λ⊗ µ . Thus, for any bounded universallymeasurable function f on E , the mapping t → Ptf(x) is Lebeague measurable, and hence, for each α ∈ [0,∞) ,one can define

Uαf(x) =

∫ ∞

0

e−αtPtf(x)dt.

There is enough joint measurability to check that Uα : α ∈ (0,∞) is a Markov resolvent on (E, E∗) , whichuniquely associated with the Markovian semigroup Pt : t ∈ [0,∞) . Consequently, one may apply Fubini’stheorem to see that

23

24 CHAPTER 11. BOREL RIGHT PROCESS

Uαf(x) = Ex

[∫ ∞

0

e−αtf(Xt)dt

].

The followings are the defining properties of Borel right processes:

• Hypothesis Droite 1:

For each probability measure µ on (E, E) , there exists a probability measure Pµ on (Ω,F∗) such that(Xt,F∗

t , Pµ) is a Markov process with initial measure µ and transition semigroup Pt : t ∈ [0,∞) .

• Hypothesis Droite 2:

Let f be α -excessive for the resolvent on (E, E∗) . Then, for each probability measure µ on (E, E) , amapping given by t→ f(Xt) is Pµ almost surely right continuous on [0,∞) .

Chapter 12

Borel set

In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, fromclosed sets) through the operations of countable union, countable intersection, and relative complement. Borel setsare named after Émile Borel.For a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra orBorel σ-algebra. The Borel algebra on X is the smallest σ-algebra containing all open sets (or, equivalently, all closedsets).Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closedsets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called aBorel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory.In some contexts, Borel sets are defined to be generated by the compact sets of the topological space, rather thanthe open sets. The two definitions are equivalent for many well-behaved spaces, including all Hausdorff σ-compactspaces, but can be different in more pathological spaces.

12.1 Generating the Borel algebra

In the case X is a metric space, the Borel algebra in the first sense may be described generatively as follows.For a collection T of subsets of X (that is, for any subset of the power set P(X) of X), let

• Tσ be all countable unions of elements of T

• Tδ be all countable intersections of elements of T

• Tδσ = (Tδ)σ.

Now define by transfinite induction a sequence Gm, where m is an ordinal number, in the following manner:

• For the base case of the definition, let G0 be the collection of open subsets of X.

• If i is not a limit ordinal, then i has an immediately preceding ordinal i − 1. Let

Gi = [Gi−1]δσ.

• If i is a limit ordinal, set

Gi =∪j<i

Gj .

The claim is that the Borel algebra isGω1 , where ω1 is the first uncountable ordinal number. That is, the Borel algebracan be generated from the class of open sets by iterating the operation

25

26 CHAPTER 12. BOREL SET

G 7→ Gδσ.

to the first uncountable ordinal.To prove this claim, note that any open set in a metric space is the union of an increasing sequence of closed sets.In particular, complementation of sets maps Gm into itself for any limit ordinal m; moreover if m is an uncountablelimit ordinal, Gm is closed under countable unions.Note that for each Borel set B, there is some countable ordinal αB such that B can be obtained by iterating theoperation over αB. However, as B varies over all Borel sets, αB will vary over all the countable ordinals, and thus thefirst ordinal at which all the Borel sets are obtained is ω1, the first uncountable ordinal.

12.1.1 Example

An important example, especially in the theory of probability, is the Borel algebra on the set of real numbers. It isthe algebra on which the Borel measure is defined. Given a real random variable defined on a probability space, itsprobability distribution is by definition also a measure on the Borel algebra.The Borel algebra on the reals is the smallest σ-algebra on R which contains all the intervals.In the construction by transfinite induction, it can be shown that, in each step, the number of sets is, at most, thepower of the continuum. So, the total number of Borel sets is less than or equal to

ℵ1 × 2ℵ0 = 2ℵ0 .

12.2 Standard Borel spaces and Kuratowski theorems

Let X be a topological space. The Borel space associated to X is the pair (X,B), where B is the σ-algebra of Borelsets of X.Mackey defined a Borel space somewhat differently, writing that it is “a set together with a distinguished σ-field ofsubsets called its Borel sets.” [1] However, modern usage is to call the distinguished sub-algebra measurable sets andsuch spaces measurable spaces. The reason for this distinction is that the Borel sets are the σ-algebra generated byopen sets (of a topological space), whereas Mackey’s definition refers to a set equipped with an arbitrary σ-algebra.There exist measurable spaces that are not Borel spaces, for any choice of topology on the underlying space.[2]

Measurable spaces form a category in which the morphisms are measurable functions between measurable spaces. Afunction f : X → Y is measurable if it pulls back measurable sets, i.e., for all measurable sets B in Y, f−1(B) is ameasurable set in X.Theorem. Let X be a Polish space, that is, a topological space such that there is a metric d on X which defines thetopology of X and which makes X a complete separable metric space. Then X as a Borel space is isomorphic to oneof (1) R, (2) Z or (3) a finite space. (This result is reminiscent of Maharam’s theorem.)Considered as Borel spaces, the real line R, the union of R with a countable set, and Rn are isomorphic.A standard Borel space is the Borel space associated to a Polish space. A standard Borel space is characterized upto isomorphism by its cardinality,[3] and any uncountable standard Borel space has the cardinality of the continuum.For subsets of Polish spaces, Borel sets can be characterized as those sets which are the ranges of continuous injectivemaps defined on Polish spaces. Note however, that the range of a continuous noninjective map may fail to be Borel.See analytic set.Every probability measure on a standard Borel space turns it into a standard probability space.

12.3 Non-Borel sets

An example of a subset of the reals which is non-Borel, due to Lusin[4] (see Sect. 62, pages 76–78), is describedbelow. In contrast, an example of a non-measurable set cannot be exhibited, though its existence can be proved.

12.4. ALTERNATIVE NON-EQUIVALENT DEFINITIONS 27

Every irrational number has a unique representation by a continued fraction

x = a0 +1

a1 +1

a2 +1

a3 +1

. . .

where a0 is some integer and all the other numbers ak are positive integers. Let A be the set of all irrationalnumbers that correspond to sequences (a0, a1, . . . ) with the following property: there exists an infinite subsequence(ak0 , ak1 , . . . ) such that each element is a divisor of the next element. This set A is not Borel. In fact, it is analytic,and complete in the class of analytic sets. For more details see descriptive set theory and the book by Kechris,especially Exercise (27.2) on page 209, Definition (22.9) on page 169, and Exercise (3.4)(ii) on page 14.Another non-Borel set is an inverse image f−1[0] of an infinite parity function f : 0, 1ω → 0, 1 . However, thisis a proof of existence (via the axiom of choice), not an explicit example.

12.4 Alternative non-equivalent definitions

According to Halmos (Halmos 1950, page 219), a subset of a locally compact Hausdorff topological space is calleda Borel set if it belongs to the smallest σ–ring containing all compact sets.Norberg and Vervaat [5] redefine the Borel algebra of a topological space X as the σ –algebra generated by its opensubsets and its compact saturated subsets. This definition is well-suited for applications in the case where X is notHausdorff. It coincides with the usual definition if X is second countable or if every compact saturated subset isclosed (which is the case in particular if X is Hausdorff).

12.5 See also

• Baire set

• Cylindrical σ-algebra

• Polish space

• Descriptive set theory

• Borel hierarchy

12.6 Notes

[1] Mackey, G.W. (1966), “Ergodic Theory and Virtual Groups”, Math. Annalen. (Springer-Verlag) 166 (3): 187–207,doi:10.1007/BF01361167, ISSN 0025-5831, (subscription required (help))

[2] Jochen Wengenroth (mathoverflow.net/users/21051), Is every sigma-algebra the Borel algebra of a topology?, http://mathoverflow.net/questions/87888 (version: 2012-02-09)

[3] Srivastava, S.M. (1991), A Course on Borel Sets, Springer Verlag, ISBN 0-387-98412-7

[4] Lusin, Nicolas (1927), “Sur les ensembles analytiques”, Fundamenta Mathematicae (Institute of mathematics, Polishacademy of sciences) 10: 1–95.

[5] Tommy Norberg and Wim Vervaat, Capacities on non-Hausdorff spaces, in: Probability and Lattices, in: CWI Tract, vol.110, Math. Centrum Centrum Wisk. Inform., Amsterdam, 1997, pp. 133-150

28 CHAPTER 12. BOREL SET

12.7 References• William Arveson, An Invitation to C*-algebras, Springer-Verlag, 1981. (See Chapter 3 for an excellent expo-sition of Polish topology)

• Richard Dudley, Real Analysis and Probability. Wadsworth, Brooks and Cole, 1989

• Halmos, Paul R. (1950). Measure theory. D. van Nostrand Co. See especially Sect. 51 “Borel sets and Bairesets”.

• Halsey Royden, Real Analysis, Prentice Hall, 1988

• Alexander S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1995 (Graduate texts in Math., vol.156)

12.8 External links• Hazewinkel, Michiel, ed. (2001), “Borel set”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Formal definition of Borel Sets in the Mizar system, and the list of theorems that have been formally provedabout it.

• Weisstein, Eric W., “Borel Set”, MathWorld.

Chapter 13

Cabal (set theory)

The Cabal was, or perhaps is, a grouping of set theorists in Southern California, particularly at UCLA and Caltech,but also at UC Irvine. Organization and procedures range from informal to nonexistent, so it is difficult to say whetherit still exists or exactly who has been a member, but it has included such notable figures as Donald A. Martin, YiannisN. Moschovakis, John R. Steel, and Alexander S. Kechris. Others who have published in the proceedings of theCabal seminar include Robert M. Solovay, W. Hugh Woodin, Matthew Foreman, and Steve Jackson.The work of the group is characterized by free use of large cardinal axioms, and research into the descriptive settheoretic behavior of sets of reals if such assumptions hold.Some of the philosophical views of the Cabal seminar were described in Maddy 1988a and Maddy 1988b.

13.1 Publications• Kechris, A. S. et al. (1978). Cabal Seminar 76-77: Proceedings. Caltech-UCLA Logic Seminar 1976-77.Springer. ISBN 0-387-09086-X.

• Kechris, A. S. (editor) (1983). Cabal Seminar 79-81: Proc Caltech-UCLA Logic Seminar 1979-81 (LectureNotes in Mathematics). Springer. ISBN 0-387-12688-0.

• Martin, D. A., A. S. Kechris, J. R. Steel (1988). Cabal Seminar 81-85: Proceedings Caltech UCLA LogicSeminar (Lecture Notes in Mathematics, No 1333). Springer. ISBN 0-387-50020-0.

• Alexander S. Kechris, Benedikt Löwe, John R. Steel (2008). Games, Scales, and Suslin cardinals: The CabalSeminar Volume I: Lecture Notes in Logic. CUP. ISBN 9780521899512.

13.2 References• Maddy, Penelope (1988). “Believing the Axioms I” (PDF). The Journal of Symbolic Logic 53 (2): 481–511.doi:10.1017/s0022481200028425.

• Maddy, Penelope (1988). “Believing the Axioms II” (PDF). The Journal of Symbolic Logic 53 (3): 736–764.doi:10.2307/2274569.

29

Chapter 14

Cantor set

In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkableand deep properties. It was discovered in 1874 by Henry John Stephen Smith[1][2][3][4] and introduced by Germanmathematician Georg Cantor in 1883.[5][6]

Through consideration of this set, Cantor and others helped lay the foundations of modern point-set topology. Al-though Cantor himself defined the set in a general, abstract way, the most commonmodern construction is theCantorternary set, built by removing the middle thirds of a line segment. Cantor himself only mentioned the ternary con-struction in passing, as an example of a more general idea, that of a perfect set that is nowhere dense.

14.1 Construction and formula of the ternary set

The Cantor ternary set is created by repeatedly deleting the open middle third of a set of line segments. One starts bydeleting the open middle third (1⁄3, 2⁄3) from the interval [0, 1], leaving two line segments: [0, 1⁄3] ∪ [2⁄3, 1]. Next,the open middle third of each of these remaining segments is deleted, leaving four line segments: [0, 1⁄9] ∪ [2⁄9, 1⁄3]∪ [2⁄3, 7⁄9] ∪ [8⁄9, 1]. This process is continued ad infinitum, where the nth set is

Cn = Cn−1

3 ∪(

23 + Cn−1

3

)and C0 = [0, 1].

The Cantor ternary set contains all points in the interval [0, 1] that are not deleted at any step in this infinite process.The first six steps of this process are illustrated below.

An explicit closed formula for the Cantor set is

C =∞∩

m=1

3m−1−1∩k=0

([0,

3k + 1

3m

]∪[3k + 2

3m, 1

])or

C = [0, 1] \∞∪

m=1

3m−1−1∪k=0

(3k + 1

3m,3k + 2

3m

).

30

14.2. COMPOSITION 31

The proof of the formula above as the special case of two family of Cantor sets is done by the idea of self-similaritytransformations and can be found in detail.[7][8]

This process of removing middle thirds is a simple example of a finite subdivision rule.It is perhaps most intuitive to think about the Cantor set as the set of real numbers between zero and one whoseternary expansion in base three doesn't contain the digit 1. This ternary digit expansion description has been more ofinterest for researchers to explore fractal and topological properties of the Cantor set.

14.2 Composition

Since the Cantor set is defined as the set of points not excluded, the proportion (i.e., measure) of the unit intervalremaining can be found by total length removed. This total is the geometric progression

∞∑n=0

2n

3n+1=

1

3+

2

9+

4

27+

8

81+ · · · = 1

3

(1

1− 23

)= 1.

So that the proportion left is 1 – 1 = 0.This calculation shows that the Cantor set cannot contain any interval of non-zero length. In fact, it may seemsurprising that there should be anything left — after all, the sum of the lengths of the removed intervals is equalto the length of the original interval. However, a closer look at the process reveals that there must be somethingleft, since removing the “middle third” of each interval involved removing open sets (sets that do not include theirendpoints). So removing the line segment (1/3, 2/3) from the original interval [0, 1] leaves behind the points 1/3 and2/3. Subsequent steps do not remove these (or other) endpoints, since the intervals removed are always internal to theintervals remaining. So the Cantor set is not empty, and in fact contains an uncountably infinite number of points.It may appear that only the endpoints are left, but that is not the case either. The number 1/4, for example, is in thebottom third, so it is not removed at the first step, and is in the top third of the bottom third, and is in the bottomthird of that, and in the top third of that, and so on ad infinitum—alternating between top and bottom thirds. Sinceit is never in one of the middle thirds, it is never removed, and yet it is also not one of the endpoints of any middlethird. The number 3/10 is also in the Cantor set and is not an endpoint.In the sense of cardinality, most members of the Cantor set are not endpoints of deleted intervals. Since each stepremoves a finite number of intervals and the number of steps is countable, the set of endpoints is countable while thewhole Cantor set is uncountable.

14.3 Properties

14.3.1 Cardinality

It can be shown that there are as many points left behind in this process as there were to begin with, and that therefore,the Cantor set is uncountable. To see this, we show that there is a function f from the Cantor set C to the closedinterval [0,1] that is surjective (i.e. f maps from C onto [0,1]) so that the cardinality of C is no less than that of [0,1].Since C is a subset of [0,1], its cardinality is also no greater, so the two cardinalities must in fact be equal, by theCantor–Bernstein–Schroeder theorem.To construct this function, consider the points in the [0, 1] interval in terms of base 3 (or ternary) notation. Recallthat some points admit more than one representation in this notation, as for example 1/3, that can be written as0.13 but also as 0.022222...3, and 2/3, that can be written as 0.23 but also as 0.12222...3. (This alternative recurringrepresentation of a number with a terminating numeral occurs in any positional system.) When we remove the middlethird, this contains the numbers with ternary numerals of the form 0.1xxxxx...3 where xxxxx...3 is strictly between00000...3 and 22222...3. So the numbers remaining after the first step consist of

• Numbers of the form 0.0xxxxx...3

• 1/3 = 0.13 = 0.022222...3

32 CHAPTER 14. CANTOR SET

• 2/3 = 0.122222...3 = 0.23• Numbers of the form 0.2xxxxx...3.

This can be summarized by saying that those numbers that admit a ternary representation such that the first digit afterthe decimal point is not 1 are the ones remaining after the first step.The second step removes numbers of the form 0.01xxxx...3 and 0.21xxxx...3, and (with appropriate care for theendpoints) it can be concluded that the remaining numbers are those with a ternary numeral where neither of the firsttwo digits is 1. Continuing in this way, for a number not to be excluded at step n, it must have a ternary representationwhose nth digit is not 1. For a number to be in the Cantor set, it must not be excluded at any step, it must admit anumeral representation consisting entirely of 0s and 2s. It is worth emphasising that numbers like 1, 1/3 = 0.13 and7/9 = 0.213 are in the Cantor set, as they have ternary numerals consisting entirely of 0s and 2s: 1 = 0.2222...3, 1/3 =0.022222...3 and 7/9 = 0.2022222...3. So while a number in C may have either a terminating or a recurring ternarynumeral, one of its representations will consist entirely of 0s and 2s.The function from C to [0,1] is defined by taking the numeral that does consist entirely of 0s and 2s, replacing all the2s by 1s, and interpreting the sequence as a binary representation of a real number. In a formula,

f

( ∞∑k=1

ak3−k

)=

∞∑k=1

ak22−k.

For any number y in [0,1], its binary representation can be translated into a ternary representation of a numberx in C by replacing all the 1s by 2s. With this, f(x) = y so that y is in the range of f. For instance if y = 3/5= 0.100110011001...2, we write x = 0.200220022002...3 = 7/10. Consequently f is surjective; however, f is notinjective — interestingly enough, the values for which f(x) coincides are those at opposing ends of one of the middlethirds removed. For instance, 7/9 = 0.2022222...3 and 8/9 = 0.2200000...3 so f(7/9) = 0.101111...2 = 0.112 = f(8/9).So there are as many points in the Cantor set as there are in [0, 1], and the Cantor set is uncountable (see Cantor’sdiagonal argument). However, the set of endpoints of the removed intervals is countable, so theremust be uncountablymany numbers in the Cantor set which are not interval endpoints. As noted above, one example of such a number is¼, which can be written as 0.02020202020...3 in ternary notation.The Cantor set contains as many points as the interval from which it is taken, yet itself contains no interval of nonzerolength. The irrational numbers have the same property, but the Cantor set has the additional property of being closed,so it is not even dense in any interval, unlike the irrational numbers which are dense in every interval.It has been conjectured that all algebraic irrational numbers are normal. Since members of the Cantor set are notnormal, this would imply that all members of the Cantor set are either rational or transcendental.

14.3.2 Self-similarity

The Cantor set is the prototype of a fractal. It is self-similar, because it is equal to two copies of itself, if each copyis shrunk by a factor of 3 and translated. More precisely, there are two functions, the left and right self-similaritytransformations, fL(x) = x/3 and fR(x) = (2+x)/3 , which leave the Cantor set invariant up to homeomorphism:fL(C) ∼= fR(C) ∼= C.

Repeated iteration of fL and fR can be visualized as an infinite binary tree. That is, at each node of the tree, onemay consider the subtree to the left or to the right. Taking the set fL, fR together with function composition formsa monoid, the dyadic monoid.The automorphisms of the binary tree are its hyperbolic rotations, and are given by the modular group. Thus, theCantor set is a homogeneous space in the sense that for any two points x and y in the Cantor set C , there existsa homeomorphism h : C → C with h(x) = y . These homeomorphisms can be expressed explicitly, as Möbiustransformations.The Hausdorff dimension of the Cantor set is equal to ln(2)/ln(3) ≈ 0.631.

14.3.3 Topological and analytical properties

Although “the” Cantor set typically refers to the original, middle-thirds Cantor desecribed above, topologists oftentalk about “a” Cantor set, which means any topological space that is homeomorphic (topologically equivalent) to it.

14.3. PROPERTIES 33

As the above summation argument shows, the Cantor set is uncountable but has Lebesgue measure 0. Since theCantor set is the complement of a union of open sets, it itself is a closed subset of the reals, and therefore a completemetric space. Since it is also totally bounded, the Heine–Borel theorem says that it must be compact.For any point in the Cantor set and any arbitrarily small neighborhood of the point, there is some other number witha ternary numeral of only 0s and 2s, as well as numbers whose ternary numerals contain 1s. Hence, every point inthe Cantor set is an accumulation point (also called a cluster point or limit point) of the Cantor set, but none is aninterior point. A closed set in which every point is an accumulation point is also called a perfect set in topology, whilea closed subset of the interval with no interior points is nowhere dense in the interval.Every point of the Cantor set is also an accumulation point of the complement of the Cantor set.For any two points in the Cantor set, there will be some ternary digit where they differ — one will have 0 and theother 2. By splitting the Cantor set into “halves” depending on the value of this digit, one obtains a partition of theCantor set into two closed sets that separate the original two points. In the relative topology on the Cantor set, thepoints have been separated by a clopen set. Consequently the Cantor set is totally disconnected. As a compact totallydisconnected Hausdorff space, the Cantor set is an example of a Stone space.As a topological space, the Cantor set is naturally homeomorphic to the product of countably many copies of thespace 0, 1 , where each copy carries the discrete topology. This is the space of all sequences in two digits

2N = (xn)|xn ∈ 0, 1 for n ∈ N

which can also be identified with the set of 2-adic integers. The basis for the open sets of the product topology arecylinder sets; the homeomorphism maps these to the subspace topology that the Cantor set inherits from the naturaltopology on the real number line. This characterization of the Cantor space as a product of compact spaces gives asecond proof that Cantor space is compact, via Tychonoff’s theorem.From the above characterization, the Cantor set is homeomorphic to the p-adic integers, and, if one point is removedfrom it, to the p-adic numbers.The Cantor set is a subset of the reals, which are a metric space with respect to the ordinary distance metric; thereforethe Cantor set itself is a metric space, by using that same metric. Alternatively, one can use the p-adic metric on 2N: given two sequences (xn), (yn) ∈ 2N , the distance between them is d((xn), (yn)) = 1/k , where k is the smallestindex such that xk = yk ; if there is no such index, then the two sequences are the same, and one defines the distanceto be zero. These two metrics generate the same topology on the Cantor set.We have seen above that the Cantor set is a totally disconnected perfect compact metric space. Indeed, in a sense itis the only one: every nonempty totally disconnected perfect compact metric space is homeomorphic to the Cantorset. See Cantor space for more on spaces homeomorphic to the Cantor set.The Cantor set is sometimes regarded as “universal” in the category of compact metric spaces, since any compactmetric space is a continuous image of the Cantor set; however this construction is not unique and so the Cantor setis not universal in the precise categorical sense. The “universal” property has important applications in functionalanalysis, where it is sometimes known as the representation theorem for compact metric spaces.[9]

For any integer q≥ 2, the topology on the groupG=Zqω (the countable direct sum) is discrete. Although the Pontrjagindual Γ is also Zqω, the topology of Γ is compact. One can see that Γ is totally disconnected and perfect - thus it ishomeomorphic to the Cantor set. It is easiest to write out the homeomorphism explicitly in the case q=2. (See Rudin1962 p 40.)

14.3.4 Measure and probability

The Cantor set can be seen as the compact group of binary sequences, and as such, it is endowed with a natural Haarmeasure. When normalized so that the measure of the set is 1, it is a model of an infinite sequence of coin tosses.Furthermore, one can show that the usual Lebesgue measure on the interval is an image of the Haar measure on theCantor set, while the natural injection into the ternary set is a canonical example of a singular measure. It can alsobe shown that the Haar measure is an image of any probability, making the Cantor set a universal probability spacein some ways.In Lebesgue measure theory, the Cantor set is an example of a set which is uncountable and has zero measure.[10]

34 CHAPTER 14. CANTOR SET

14.4 Variants

14.4.1 Smith–Volterra–Cantor set

Main article: Smith–Volterra–Cantor set

Instead of repeatedly removing the middle third of every piece as in the Cantor set, we could also keep removing anyother fixed percentage (other than 0% and 100%) from the middle. In the case where the middle 8/10 of the intervalis removed, we get a remarkably accessible case — the set consists of all numbers in [0,1] that can be written as adecimal consisting entirely of 0s and 9s.By removing progressively smaller percentages of the remaining pieces in every step, one can also construct setshomeomorphic to the Cantor set that have positive Lebesgue measure, while still being nowhere dense. See Smith–Volterra–Cantor set for an example.

14.4.2 Cantor dust

Cantor dust is a multi-dimensional version of the Cantor set. It can be formed by taking a finite Cartesian productof the Cantor set with itself, making it a Cantor space. Like the Cantor set, Cantor dust has zero measure.[11]

A different 2D analogue of the Cantor set is the Sierpinski carpet, where a square is divided up into nine smallersquares, and the middle one removed. The remaining squares are then further divided into nine each and the middleremoved, and so on ad infinitum.[12] The 3D analogue of this is the Menger sponge.

14.5 Historical remarks

Cantor himself defined the set in a general, abstract way, and mentioned the ternary construction only in passing, asan example of a more general idea, that of a perfect set that is nowhere dense. The original paper provides severaldifferent constructions of the abstract concept.This set would have been considered abstract at the time when Cantor devised it. Cantor himself was led to it bypractical concerns about the set of points where a trigonometric series might fail to converge. The discovery did muchto set him on the course for developing an abstract, general theory of infinite sets.

14.6 See also• Cantor function• Cantor cube• Antoine’s necklace• Koch snowflake• Knaster–Kuratowski fan• List of fractals by Hausdorff dimension

14.7 Notes[1] Henry J.S. Smith (1874) “On the integration of discontinuous functions.” Proceedings of the London Mathematical Society,

Series 1, vol. 6, pages 140–153.

[2] The “Cantor set” was also discovered by Paul du Bois-Reymond (1831–1889). See footnote on page 128 of: Paul duBois-Reymond (1880) “Der Beweis des Fundamentalsatzes der Integralrechnung,”Mathematische Annalen, vol. 16, pages115–128. The “Cantor set” was also discovered in 1881 by Vito Volterra (1860–1940). See: Vito Volterra (1881) “Alcuneosservazioni sulle funzioni punteggiate discontinue” [Some observations on point-wise discontinuous functions], Giornaledi Matematiche, vol. 19, pages 76–86.

14.8. REFERENCES 35

[3] José Ferreirós, Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics (Basel, Switzerland:Birkhäuser Verlag, 1999), pages 162–165.

[4] Ian Stewart, Does God Play Dice?: The New Mathematics of Chaos

[5] Georg Cantor (1883) "Über unendliche, lineare Punktmannigfaltigkeiten V" [On infinite, linear point-manifolds (sets)],Mathematische Annalen, vol. 21, pages 545–591.

[6] H.-O. Peitgen, H. Jürgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science 2nd ed. (N.Y., N.Y.: SpringerVerlag, 2004), page 65.

[7] Mohsen Soltanifar, On A sequence of cantor Fractals, Rose Hulman Undergraduate Mathematics Journal, Vol 7, No 1,paper 9, 2006.

[8] Mohsen Soltanifar, A Different Description of A Family of Middle-a Cantor Sets, American Journal of UndergraduateResearch, Vol 5, No 2, pp 9–12, 2006.

[9] Stephen Willard, General Topology, Addison-Wesley Publishing Company, 1968.

[10] the Cantor set is an uncountable set with zero measure

[11] Helmberg, Gilbert (2007). Getting Acquainted With Fractals. Walter de Gruyter. p. 46. ISBN 978-3-11-019092-2.

[12] Helmberg, Gilbert (2007). Getting Acquainted With Fractals. Walter de Gruyter. p. 48. ISBN 978-3-11-019092-2.

14.8 References• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446 (See example 29).

• Gary L. Wise and Eric B. Hall, Counterexamples in Probability and Real Analysis. Oxford University Press,New York 1993. ISBN 0-19-507068-2. (See chapter 1).

14.9 External links• Hazewinkel, Michiel, ed. (2001), “Cantor set”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Cantor Sets and Cantor Set and Function at cut-the-knot

• Cantor Set (PRIME)

• Cantor Dust Demo Program

Chapter 15

Cantor space

In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: atopological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ωis called “the” Cantor space. Note that, commonly, 2ω is referred to simply as the Cantor set, while the term Cantorspace is reserved for the more general construction ofDS for a finite setD and a set S which might be finite, countableor possibly uncountable.[1]

15.1 Examples

The Cantor set itself is a Cantor space. But the canonical example of a Cantor space is the countably infinitetopological product of the discrete 2-point space 0, 1. This is usually written as 2N or 2ω (where 2 denotes the2-element set 0,1 with the discrete topology). A point in 2ω is an infinite binary sequence, that is a sequence whichassumes only the values 0 or 1. Given such a sequence a0, a1, a2,..., one can map it to the real number

∞∑n=0

2an3n+1

.

This mapping gives a homeomorphism from 2ω onto the Cantor set, demonstrating that 2ω is indeed a Cantor space.Cantor spaces occur abundantly in real analysis. For example, they exist as subspaces in every perfect, completemetric space. (To see this, note that in such a space, any non-empty perfect set contains two disjoint non-emptyperfect subsets of arbitrarily small diameter, and so one can imitate the construction of the usual Cantor set.) Also,every uncountable, separable, completely metrizable space contains Cantor spaces as subspaces. This includes mostof the common type of spaces in real analysis.

15.2 Characterization

A topological characterization of Cantor spaces is given by Brouwer's theorem:[2]

Any two non-empty compact Hausdorff spaces without isolated points and having countable bases consist-ing of clopen sets are homeomorphic to each other.

The topological property of having a base consisting of clopen sets is sometimes known as “zero-dimensionality”.Brouwer’s theorem can be restated as:

A topological space is a Cantor space if and only if it is non-empty, perfect, compact, totally disconnected,and metrizable.

This theorem is also equivalent (via Stone’s representation theorem for Boolean algebras) to the fact that any twocountable atomless Boolean algebras are isomorphic.

36

15.3. PROPERTIES 37

15.3 Properties

As can be expected from Brouwer’s theorem, Cantor spaces appear in several forms. But many properties of Cantorspaces can be established using 2ω, because its construction as a product makes it amenable to analysis.Cantor spaces have the following properties:

• The cardinality of any Cantor space is 2ℵ0 , that is, the cardinality of the continuum.

• The product of two (or even any finite or countable number of) Cantor spaces is a Cantor space. Along withthe Cantor function; this fact can be used to construct space-filling curves.

• A Hausdorff topological space is compact metrizable if and only if it is a continuous image of a Cantorspace.[3][4]

Let C(X) denote the space of all real-valued, bounded continuous functions on a topological space X. Let K denote acompact metric space, and Δ denote the Cantor set. Then the Cantor set has the following property:

• C(K) is isometric to a closed subspace of C(Δ).[5]

In general, this isometry is not unique, and thus is not properly a universal property in the categorical sense.

• The group of all homeomorphisms of the Cantor space is simple.[6]

15.4 See also• Space (mathematics)

• Cantor set

• Cantor cube

15.5 References[1] Stephen Willard, General Topology (1970) Addison-Wesley Publishing. See section 17.9a

[2] Brouwer, L. E. J. (1910), “On the structure of perfect sets of points” (PDF), Proc. Koninklijke Akademie vanWetenschappen12: 785–794.

[3] N.L. Carothers, A Short Course on Banach Space Theory, London Mathematical Society Student Texts 64, (2005) Cam-bridge University Press. See Chapter 12

[4] Willard, op.cit., See section 30.7

[5] Carothers, op.cit.

[6] R.D. Anderson, The Algebraic Simplicity of Certain Groups of Homeomorphisms, American Journal of Mathematics 80(1958), pp. 955-963.

• Kechris, A. (1995). Classical Descriptive Set Theory (Graduate Texts inMathematics 156 ed.). Springer. ISBN0-387-94374-9.

Chapter 16

Choquet game

In mathematics, a Choquet game, introduced by Gustave Choquet (1969), is a topological game where two playerstake turns decreasing the size of a non-empty open subset of a topological space, and the first player wins if after aninfinite number of moves the open sets have empty intersection. A nonempty topological space where the secondplayer has a winning strategy is called a Choquet space.A nonempty topological space where the first player has no winning strategy is the same as a Baire space, so inparticular every Choquet space is a Baire space. However there are separable metric spaces where neither player hasa winning strategy, so there are Baire spaces that are not Choquet spaces. Every nonempty complete metric space ornonempty locally compact Hausdorff space is a Choquet space.

16.1 References• Choquet, Gustave (1969), Lectures on analysis. Vol. I: Integration and topological vector spaces, New York-Amsterdam: W. A. Benjamin, Inc., MR 0250011

• Kechris, Alexander S. (1994). Classical Descriptive Set Theory. Springer-Verlag. ISBN 0-387-94374-9.

38

Chapter 17

Cichoń's diagram

In set theory, Cichoń's diagram or Cichon’s diagram is a table of 10 infinite cardinal numbers related to the settheory of the reals displaying the provable relations between these cardinal characteristics of the continuum. All thesecardinals are greater than or equal to ℵ1 , the smallest uncountable cardinal, and they are bounded above by 2ℵ0 ,the cardinality of the continuum. Four cardinals describe properties of the ideal of sets of measure zero; four moredescribe the corresponding properties of the ideal of meager sets (first category sets).

17.1 Definitions

Let I be an ideal of a fixed infinite set X, containing all finite subsets of X. We define the following "cardinal coeffi-cients" of I:

• add(I) = min|A| : A ⊆ I ∧∪A /∈ I

.

The “additivity” of I is the smallest number of sets from I whose union is not in I any more.As any ideal is closed under finite unions, this number is always at least ℵ0 ; if I is a σ-ideal,then add(I)≥ ℵ1 .

• cov(I) = min|A| : A ⊆ I ∧∪A = X

.

The “covering number” of I is the smallest number of sets from I whose union is all of X. AsX itself is not in I, we must have add(I) ≤ cov(I).

• non(I) = min|A| : A ⊆ X ∧ A /∈ I,

The “uniformity number” of I (sometimes also written unif(I) ) is the size of the smallest setnot in I. By our assumption on I, add(I) ≤ non(I).

• cof(I) = min|B| : B ⊆ I ∧ (∀A ∈ I)(∃B ∈ B)(A ⊆ B).

The “cofinality” of I is the cofinality of the partial order (I, ⊆). It is easy to see that we musthave non(I) ≤ cof(I) and cov(I) ≤ cof(I).

Furthermore, the "bounding number" or “unboundedness number” b and the "dominating number" d are defined asfollows:

• b = min|F | : F ⊆ NN ∧ (∀g ∈ NN)(∃f ∈ F )(∃∞n ∈ N)(g(n) < f(n))

,

• d = min|F | : F ⊆ NN ∧ (∀g ∈ NN)(∃f ∈ F )(∀∞n ∈ N)(g(n) < f(n))

,

where " ∃∞n ∈ N " means: “there are infinitely many natural numbers n such that...”, and " ∀∞n ∈ N " means “forall except finitely many natural numbers n we have...”.

39

40 CHAPTER 17. CICHOŃ'S DIAGRAM

17.2 Diagram

Let K be the σ-ideal of those subsets of the real line which are meager (or “of the first category”) in the euclideantopology, and let L be the σ-ideal of those subsets of the real line which are of Lebesgue measure zero. Then thefollowing inequalities hold (where an arrow from a to b is to be read as meaning that a ≤ b):

In addition, the following relations hold:

add(K) = mincov(K), b and cof(K) = maxnon(K), d .[1]

It turns out that the inequalities described by the diagram, together with the relations mentioned above, are all therelations between these cardinals that are provable in ZFC, in the following sense. Let A be any assignment of thecardinals ℵ1 and ℵ2 to the 10 cardinals in Cichoń's diagram. Then, if A is consistent with the diagram in that thereis no arrow from ℵ2 to ℵ1 , and if A also satisfies the two additional relations, then A can be realized in some modelof ZFC.Some inequalities in the diagram (such as “add ≤ cov”) follow immediately from the definitions. The inequalitiescov(K) ≤ non(L) and cov(L) ≤ non(K) are classical theorems and follow from the fact that the real line can bepartitioned into a meager set and a set of measure zero.

17.3 Remarks

The British mathematician David Fremlin named the diagram after the Wrocław mathematician Jacek Cichoń.[2]

The continuum hypothesis, of 2ℵ0 being equal to ℵ1 , would make all of these arrows equalities.Martin’s axiom, a weakening of CH, implies that all cardinals in the diagram (except perhaps ℵ1 ) are equal to 2ℵ0 .

17.4 References[1] Bartoszyński, Tomek (2009), “Invariants of Measure and Category”, in Foreman, Matthew, Handbook of Set Theory,

Springer-Verlag, pp. 491–555, arXiv:math/9910015, doi:10.1007/978-1-4020-5764-9_8, ISBN 978-1-4020-4843-2

[2] Fremlin, David H. (1984), “Cichon’s diagram”, Sémin. Initiation Anal. 23ème Année-1983/84, Publ. Math. Pierre andMarie Curie University 66, Zbl 0559.03029, Exp. No.5, 13 p.,.

Chapter 18

Coanalytic set

In the mathematical discipline of descriptive set theory, a coanalytic set is a set (typically a set of real numbers ormore generally a subset of a Polish space) that is the complement of an analytic set (Kechris 1994:87). Coanalyticsets are also referred to as Π1

1 sets (see projective hierarchy).

18.1 References• Kechris, Alexander S. (1994), Classical Descriptive Set Theory, Springer-Verlag, ISBN 0-387-94374-9

41

Chapter 19

Descriptive set theory

In mathematical logic, descriptive set theory is the study of certain classes of "well-behaved" subsets of the realline and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applicationsto other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and groupactions, and mathematical logic.

19.1 Polish spaces

Descriptive set theory begins with the study of Polish spaces and their Borel sets.A Polish space is a second countable topological space that is metrizable with a complete metric. Equivalently, it isa complete separable metric space whose metric has been “forgotten”. Examples include the real line R , the Bairespace N , the Cantor space C , and the Hilbert cube IN .

19.1.1 Universality properties

The class of Polish spaces has several universality properties, which show that there is no loss of generality in con-sidering Polish spaces of certain restricted forms.

• Every Polish space is homeomorphic to a Gδ subspace of the Hilbert cube, and every Gδ subspace of theHilbert cube is Polish.

• Every Polish space is obtained as a continuous image of Baire space; in fact every Polish space is the imageof a continuous bijection defined on a closed subset of Baire space. Similarly, every compact Polish space is acontinuous image of Cantor space.

Because of these universality properties, and because the Baire space N has the convenient property that it ishomeomorphic to Nω , many results in descriptive set theory are proved in the context of Baire space alone.

19.2 Borel sets

The class of Borel sets of a topological space X consists of all sets in the smallest σ-algebra containing the open setsof X. This means that the Borel sets of X are the smallest collection of sets such that:

• Every open subset of X is a Borel set.

• If A is a Borel set, so is X \A . That is, the class of Borel sets are closed under complementation.

• If An is a Borel set for each natural number n, then the union∪An is a Borel set. That is, the Borel sets are

closed under countable unions.

42

19.3. ANALYTIC AND COANALYTIC SETS 43

A fundamental result shows that any two uncountable Polish spaces X and Y are Borel isomorphic: there is a bijectionfrom X to Y such that the preimage of any Borel set is Borel, and the image of any Borel set is Borel. This givesadditional justification to the practice of restricting attention to Baire space and Cantor space, since these and anyother Polish spaces are all isomorphic at the level of Borel sets.

19.2.1 Borel hierarchy

Each Borel set of a Polish space is classified in the Borel hierarchy based on how many times the operations ofcountable union and complementation must be used to obtain the set, beginning from open sets. The classification isin terms of countable ordinal numbers. For each nonzero countable ordinal α there are classes 0

α , 0α , and 0

α .

• Every open set is declared to be 01 .

• A set is declared to be 0α if and only if its complement is 0

α .

• A set A is declared to be 0δ , δ > 1, if there is a sequence ⟨ Ai ⟩ of sets, each of which is 0

λ(i) for some λ(i) <δ, such that A =

∪Ai .

• A set is 0α if and only if it is both 0

α and 0α .

A theorem shows that any set that is 0α or 0

α is 0α+1 , and any 0

β set is both 0α and 0

α for all α > β. Thus thehierarchy has the following structure, where arrows indicate inclusion.

01

02 · · ·

01

02 · · ·

01

02 · · ·

0α · · ·

0α

0α+1 · · ·

0α · · ·

19.2.2 Regularity properties of Borel sets

Classical descriptive set theory includes the study of regularity properties of Borel sets. For example, all Borel setsof a Polish space have the property of Baire and the perfect set property. Modern descriptive set theory includes thestudy of the ways in which these results generalize, or fail to generalize, to other classes of subsets of Polish spaces.

19.3 Analytic and coanalytic sets

Just beyond the Borel sets in complexity are the analytic sets and coanalytic sets. A subset of a Polish space X isanalytic if it is the continuous image of a Borel subset of some other Polish space. Although any continuous preimageof a Borel set is Borel, not all analytic sets are Borel sets. A set is coanalytic if its complement is analytic.

19.4 Projective sets and Wadge degrees

Many questions in descriptive set theory ultimately depend upon set-theoretic considerations and the properties ofordinal and cardinal numbers. This phenomenon is particularly apparent in the projective sets. These are definedvia the projective hierarchy on a Polish space X:

• A set is declared to be 11 if it is analytic.

• A set is 11 if it is coanalytic.

• A set A is 1n+1 if there is a 1

n subset B of X ×X such that A is the projection of B to the first coordinate.

• A set A is 1n+1 if there is a 1

n subset B of X ×X such that A is the projection of B to the first coordinate.

• A set is 1n if it is both 1

n and 1n .

44 CHAPTER 19. DESCRIPTIVE SET THEORY

As with the Borel hierarchy, for each n, any 1n set is both 1

n+1 and 1n+1.

The properties of the projective sets are not completely determined by ZFC. Under the assumption V = L, not allprojective sets have the perfect set property or the property of Baire. However, under the assumption of projectivedeterminacy, all projective sets have both the perfect set property and the property of Baire. This is related to thefact that ZFC proves Borel determinacy, but not projective determinacy.More generally, the entire collection of sets of elements of a Polish space X can be grouped into equivalence classes,known asWadge degrees, that generalize the projective hierarchy. These degrees are ordered in theWadge hierarchy.The axiom of determinacy implies that the Wadge hierarchy on any Polish space is well-founded and of length Θ,with structure extending the projective hierarchy.

19.5 Borel equivalence relations

A contemporary area of research in descriptive set theory studies Borel equivalence relations. A Borel equivalencerelation on a Polish space X is a Borel subset of X ×X that is an equivalence relation on X.

19.6 Effective descriptive set theory

The area of effective descriptive set theory combines the methods of descriptive set theory with those of generalizedrecursion theory (especially hyperarithmetical theory). In particular, it focuses on lightface analogues of hierarchiesof classical descriptive set theory. Thus the hyperarithmetic hierarchy is studied instead of the Borel hierarchy, andthe analytical hierarchy instead of the projective hierarchy. This research is related to weaker version of set theorysuch as Kripke-Platek set theory and second-order arithmetic.

19.7 See also• Pointclass

• Prewellordering

• Scale property

19.8 References• Kechris, Alexander S. (1994). Classical Descriptive Set Theory. Springer-Verlag. ISBN 0-387-94374-9.

• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0. Secondedition available online

19.9 External links• Descriptive set theory, David Marker, 2002. Lecture notes.

Chapter 20

Difference hierarchy

In set theory, the difference hierarchy over a pointclass is a hierarchy of larger pointclasses generated by takingdifferences of sets. If Γ is a pointclass, then the set of differences in Γ is A : ∃C,D ∈ Γ(A = C \D) . In usualnotation, this set is denoted by 2-Γ. The next level of the hierarchy is denoted by 3-Γ and consists of differences ofthree sets: A : ∃C,D,E ∈ Γ(A = C \ (D \E)) . This definition can be extended recursively into the transfiniteto α-Γ for some ordinal α.[1]

In the Borel and projective hierarchies, Felix Hausdorff proved that the countable levels of the difference hierarchyover Π0ᵧ and Π1ᵧ give Δ0ᵧ₊₁ and Δ1ᵧ₊₁, respectively.[2]

20.1 References[1] Kanamori, Akihiro (2009), The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, SpringerMonographs

in Mathematics (2nd ed.), Springer-Verlag, Berlin, p. 442, ISBN 978-3-540-88866-6, MR 2731169.

[2] Wadge, WilliamW. (2012), “Early investigations of the degrees of Borel sets”,Wadge degrees and projective ordinals. TheCabal Seminar. Volume II, Lect. Notes Log. 37, Assoc. Symbol. Logic, La Jolla, CA, pp. 166–195, MR 2906999. Seein particular p. 173.

45

Chapter 21

Effective descriptive set theory

Effective descriptive set theory is the branch of descriptive set theory dealing with sets of reals having lightfacedefinitions; that is, definitions that do not require an arbitrary real parameter (Moschovakis 1980). Thus effectivedescriptive set theory combines descriptive set theory with recursion theory.

21.1 Constructions

21.1.1 Effective Polish space

Main article: Effective Polish space

An effective Polish space is a complete separable metric space that has a computable presentation. Such spaces arestudied in both effective descriptive set theory and in constructive analysis. In particular, standard examples of Polishspaces such as the real line, the Cantor set and the Baire space are all effective Polish spaces.

21.1.2 Arithmetical hierarchy

Main article: Arithmetical hierarchy

The arithmetical hierarchy, arithmetic hierarchy or Kleene-Mostowski hierarchy classifies certain sets based onthe complexity of formulas that define them. Any set that receives a classification is called arithmetical.More formally, the arithmetical hierarchy assigns classifications to the formulas in the language of first-order arith-metic. The classifications are denoted Σ0

n and Π0n for natural numbers n (including 0). The Greek letters here are

lightface symbols, which indicates that the formulas do not contain set parameters.If a formula ϕ is logically equivalent to a formula with only bounded quantifiers then ϕ is assigned the classificationsΣ0

0 and Π00 .

The classifications Σ0n and Π0

n are defined inductively for every natural number n using the following rules:

• If ϕ is logically equivalent to a formula of the form ∃n1∃n2 · · · ∃nkψ , where ψ is Π0n , then ϕ is assigned the

classification Σ0n+1 .

• If ϕ is logically equivalent to a formula of the form ∀n1∀n2 · · · ∀nkψ , where ψ is Σ0n , then ϕ is assigned the

classification Π0n+1 .

21.2 References• Mansfield, Richard; Weitkamp, Galen (1985). Recursive Aspects of Descriptive Set Theory. Oxford UniversityPress. pp. 124–38. ISBN 978-0-19-503602-2. MR 786122.

46

21.2. REFERENCES 47

• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0. Secondedition available online

Chapter 22

Effective Polish space

In mathematical logic, an effective Polish space is a complete separable metric space that has a computable presen-tation. Such spaces are studied in effective descriptive set theory and in constructive analysis. In particular, standardexamples of Polish spaces such as the real line, the Cantor set and the Baire space are all effective Polish spaces.

22.1 Definition

An effective Polish space is a complete separable metric space X with metric d such that there is a countable denseset C = (c0, c1,...) that makes the following two relations on N4 computable (Moschovakis 2009:96-7):

P (i, j, k,m) ≡ d(ci, cj) ≤m

k + 1

Q(i, j, k,m) ≡ d(ci, cj) <m

k + 1

22.2 References• Yiannis N. Moschovakis, 2009, Descriptive Set Theory, 2nd edition, American Mathematical Society. ISBN0-8218-4813-5

48

Chapter 23

Fσ set

In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets. The notation originated in Francewith F for fermé (French: closed) and σ for somme (French: sum, union).[1]

In metrizable spaces, every open set is an Fσ set.[2] The complement of an Fσ set is a Gδ set.[1] In a metrizable space,any closed set is a Gδ set.The union of countably many Fσ sets is an Fσ set, and the intersection of finitely many Fσ sets is an Fσ set. Fσ is thesame as 0

2 in the Borel hierarchy.

23.1 Examples

Each closed set is an Fσ set.The set Q of rationals is an Fσ set. The set R \Q of irrationals is not a Fσ set.In a Tychonoff space, each countable set is an Fσ set, because a point x is closed.For example, the set A of all points (x, y) in the Cartesian plane such that x/y is rational is an Fσ set because it canbe expressed as the union of all the lines passing through the origin with rational slope:

A =∪r∈Q

(ry, y) | y ∈ R,

where Q , is the set of rational numbers, which is a countable set.

23.2 See also• Gδ set — the dual notion.

• Borel hierarchy

• P-space, any space having the property that every Fσ set is closed

23.3 References[1] Stein, Elias M.; Shakarchi, Rami (2009), Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton Uni-

versity Press, p. 23, ISBN 9781400835560.

[2] Aliprantis, Charalambos D.; Border, Kim (2006), Infinite Dimensional Analysis: A Hitchhiker’s Guide, Springer, p. 138,ISBN 9783540295877.

49

Chapter 24

Gregory number

In mathematics, a Gregory number, named after James Gregory, is a real number of the form:[1]

Gx =∞∑i=0

(−1)i1

(2i+ 1)x2i+1

where x is any rational number greater or equal to 1. Considering the power series expansion for arctangent, we have

Gx = arctan 1

x.

Setting x = 1 gives the well-known Leibniz formula for pi.

24.1 See also• Størmer number

24.2 References[1] Conway, John H.; R. K. Guy (1996). The Book of Numbers. New York: Copernicus Press. pp. 241–243.

50

Chapter 25

Gδ set

In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection ofopen sets. The notation originated in Germany with G for Gebiet (German: area, or neighborhood) meaning open setin this case and δ for Durchschnitt (German: intersection). The term inner limiting set is also used. Gδ sets, andtheir dual Fσ sets, are the second level of the Borel hierarchy.

25.1 Definition

In a topological space a Gδ set is a countable intersection of open sets. The Gδ sets are exactly the level 02 sets of

the Borel hierarchy.

25.2 Examples• Any open set is trivially a Gδ set

• The irrational numbers are a Gδ set in the real numbers R. They can be written as the countable intersectionof the sets qC where q is rational.

• The set of rational numbers Q is not a Gδ set in R. If Q were the intersection of open sets An, each An wouldbe dense in R because Q is dense in R. However, the construction above gave the irrational numbers as acountable intersection of open dense subsets. Taking the intersection of both of these sets gives the empty setas a countable intersection of open dense sets in R, a violation of the Baire category theorem.

• The zero-set of a derivative of an everywhere differentiable real-valued function on R is a Gδ set; it can be adense set with empty interior, as shown by Pompeiu’s construction.

A more elaborate example of a Gδ set is given by the following theorem:Theorem: The setD = f ∈ C([0, 1]) : f of point any at differentiable not is [0, 1] contains a dense Gδ subset ofthe metric space C([0, 1]) . (See Weierstrass function#Density of nowhere-differentiable functions.)

25.3 Properties

The notion of Gδ sets in metric (and topological) spaces is strongly related to the notion of completeness of the metricspace as well as to the Baire category theorem. This is described by the Mazurkiewicz theorem:Theorem (Mazurkiewicz): Let (X , ρ) be a complete metric space and A ⊂ X . Then the following are equivalent:

1. A is a Gδ subset of X

51

52 CHAPTER 25. GΔ SET

2. There is a metric σ on A which is equivalent to ρ|A such that (A, σ) is a complete metric space.

A key property of Gδ sets is that they are the possible sets at which a function from a topological space to a metricspace is continuous. Formally: The set of points where a function f is continuous is a Gδ set. This is becausecontinuity at a point p can be defined by a Π0

2 formula, namely: For all positive integers n , there is an open set Ucontaining p such that d(f(x), f(y)) < 1/n for all x, y in U . If a value of n is fixed, the set of p for which thereis such a corresponding open U is itself an open set (being a union of open sets), and the universal quantifier on ncorresponds to the (countable) intersection of these sets. In the real line, the converse holds as well; for any Gδ subsetA of the real line, there is a function f: R → R which is continuous exactly at the points in A. As a consequence,while it is possible for the irrationals to be the set of continuity points of a function (see the popcorn function), it isimpossible to construct a function which is continuous only on the rational numbers.

25.3.1 Basic properties

• The complement of a Gδ set is an Fσ set.

• The intersection of countably many Gδ sets is a Gδ set, and the union of finitely many Gδ sets is a Gδ set; acountable union of Gδ sets is called a Gδσ set.

• In metrizable spaces, every closed set is a Gδ set and, dually, every open set is an Fσ set.

• A subspace A of a completely metrizable space X is itself completely metrizable if and only if A is a Gδ set inX.

• A set that contains the intersection of a countable collection of dense open sets is called comeagre or residual.These sets are used to define generic properties of topological spaces of functions.

The following results regard Polish spaces:[1]

• Let (X , T ) be a Polish topological space and let G ⊂ X be a Gδ set (with respect to T ). Then G is a Polishspace with respect to the subspace topology on it.

• Topological characterization of Polish spaces: If X is a Polish space then it is homeomorphic to a Gδ subsetof a compact metric space.

25.4 Gδ space

A Gδ space is a topological space in which every closed set is a Gδ set (Johnson 1970). A normal space which isalso a Gδ space is perfectly normal. Every metrizable space is perfectly normal, and every perfectly normal spaceis completely normal: neither implication is reversible.

25.5 See also• Fσ set, the dual concept; note that “G” is German (Gebiet) and “F” is French (fermé).

• P-space, any space having the property that every Gδ set is open

25.6 References• Kelley, John L. (1955). General topology. van Nostrand. p. 134.

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 507446 P. 162.

25.7. NOTES 53

• Fremlin, D.H. (2003) [2003]. “4, General Topology”. Measure Theory, Volume 4. Petersburg, England:Digital Books Logostics. ISBN 0-9538129-4-4. Retrieved 1 April 2011 P. 334.

• Johnson, Roy A. (1970). “A Compact Non-Metrizable Space Such That Every Closed Subset is a G-Delta”.The American Mathematical Monthly 77 (2): 172–176. JSTOR 2317335.

25.7 Notes[1] Fremlin, D.H. (2003). “4, General Topology”. Measure Theory, Volume 4. Petersburg, England: Digital Books Logistics.

pp. 334–335. ISBN 0-9538129-4-4. Retrieved 1 April 2011.

Chapter 26

Homogeneous tree

In descriptive set theory, a tree over a product set Y ×Z is said to be homogeneous if there is a system of measures⟨µs | s ∈ <ωY ⟩ such that the following conditions hold:

• µs is a countably-additive measure on t | ⟨s, t⟩ ∈ T .

• The measures are in some sense compatible under restriction of sequences: if s1 ⊆ s2 , then µs1(X) =1 ⇐⇒ µs2(t | t lh(s1) ∈ X) = 1 .

• If x is in the projection of T , the ultrapower by ⟨µxn | n ∈ ω⟩ is wellfounded.

An equivalent definition is produced when the final condition is replaced with the following:

• There are ⟨µs | s ∈ ωY ⟩ such that if x is in the projection of [T ] and ∀n ∈ ω µxn(Xn) = 1 , then there isf ∈ ωZ such that ∀n ∈ ω f n ∈ Xn . This condition can be thought of as a sort of countable completenesscondition on the system of measures.

T is said to be κ -homogeneous if each µs is κ -complete.Homogeneous trees are involved in Martin and Steel's proof of projective determinacy.

26.1 References• Martin, Donald A. and John R. Steel (Jan 1989). “A Proof of Projective Determinacy”. Journal of the Amer-

ican Mathematical Society (Journal of the American Mathematical Society, Vol. 2, No. 1) 2 (1): 71–125.doi:10.2307/1990913. JSTOR 1990913.

54

Chapter 27

Homogeneously Suslin set

In descriptive set theory, a set S is said to be homogeneously Suslin if it is the projection of a homogeneous tree. Sis said to be κ -homogeneously Suslin if it is the projection of a κ -homogeneous tree.If A ⊆ ωω is a 1

1 set and κ is a measurable cardinal, then A is κ -homogeneously Suslin. This result is important inthe proof that the existence of a measurable cardinal implies that 1

1 sets are determined.

27.1 See also• Projective determinacy

27.2 References• Martin, Donald A. and John R. Steel (Jan 1989). “A Proof of Projective Determinacy”. Journal of the Amer-

ican Mathematical Society (American Mathematical Society) 2 (1): 71–125. doi:10.2307/1990913. JSTOR1990913.

55

Chapter 28

Inductive set

This article is about the notion in descriptive set theory. For the use in foundations of mathematics, see axiom ofinfinity.

Bourbaki also defines an inductive set to be a partially ordered set that satisfies the hypothesis of Zorn’slemma when nonempty.

In descriptive set theory, an inductive set of real numbers (or more generally, an inductive subset of a Polish space)is one that can be defined as the least fixed point of a monotone operation definable by a positive Σ1n formula, forsome natural number n, together with a real parameter.The inductive sets form a boldface pointclass; that is, they are closed under continuous preimages. In the Wadgehierarchy, they lie above the projective sets and below the sets in L(R). Assuming sufficient determinacy, the class ofinductive sets has the scale property and thus the prewellordering property.

28.1 References• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

56

Chapter 29

Infinity-Borel set

In set theory, a subset of a Polish space X is ∞-Borel if it can be obtained by starting with the open subsets of X, and transfinitely iterating the operations of complementation and wellordered union. Note that the set of ∞-Borelsets may not actually be closed under wellordered union; see below.

29.1 Formal definition

More formally: we define by simultaneous transfinite recursion the notion of∞-Borel code, and of the interpretationof such codes. Since X is Polish, it has a countable base. Let ⟨Ni|i < ω⟩ enumerate that base (that is, Ni is the ithbasic open set). Now:

• Every natural number i is an ∞-Borel code. Its interpretation is Ni .

• If c is an ∞-Borel code with interpretation Ac , then the ordered pair ⟨0, c⟩ is also an ∞-Borel code, and itsinterpretation is the complement of Ac , that is, X \Ac .

• If c is a length-α sequence of ∞-Borel codes for some ordinal α (that is, if for every β<α, cβ is an ∞-Borelcode, say with interpretation Acβ ), then the ordered pair ⟨1, c⟩ is an ∞-Borel code, and its interpretation is∪

β<αAcβ .

Now a set is ∞-Borel if it is the interpretation of some ∞-Borel code.The axiom of choice implies that every set can be wellordered, and therefore that every subset of every Polish spaceis ∞-Borel. Therefore the notion is interesting only in contexts where AC does not hold (or is not known to hold).Unfortunately, without the axiom of choice, it is not clear that the ∞-Borel sets are closed under wellordered union.This is because, given a wellordered union of ∞-Borel sets, each of the individual sets may havemany∞-Borel codes,and there may be no way to choose one code for each of the sets, with which to form the code for the union.The assumption that every set of reals is ∞-Borel is part of AD+, an extension of the axiom of determinacy studiedby Woodin.

29.2 Incorrect definition

It is very tempting to read the informal description at the top of this article as claiming that the ∞-Borel sets are thesmallest class of subsets ofX containing all the open sets and closed under complementation and wellordered union.That is, one might wish to dispense with the ∞-Borel codes altogether and try a definition like this:

For each ordinal α define by transfinite recursion Bα as follows:

1. B0 is the collection of all open subsets of X .2. For a given even ordinal α, Bα₊₁ is the union of Bα with the set of all complements of sets in Bα.

57

58 CHAPTER 29. INFINITY-BOREL SET

3. For a given even ordinal α, Bα₊₂ is the set of all wellordered unions of sets in Bα₊₁.4. For a given limit ordinal λ, Bλ is the union of all Bα for α<λ

It follows from the Burali-Forti paradox that there must be some ordinal α such that Bᵦ equals Bα forevery β>α. For this value of α, Bα is the collection of "∞-Borel sets”.

This set is manifestly closed under well-ordered unions, but without AC it cannot be proved equal to the ∞-Borel sets(as defined in the previous section). Specifically, it is instead the closure of the ∞-Borel sets under all well-orderedunions, even those for which a choice of codes cannot be made.

29.3 Alternative characterization

For subsets of Baire space or Cantor space, there is a more concise (if less transparent) alternative definition, whichturns out to be equivalent. A subset A of Baire space is ∞-Borel just in case there is a set of ordinals S and a first-orderformula φ of the language of set theory such that, for every x in Baire space,

x ∈ A ⇐⇒ L[S, x] |= ϕ(S, x)

where L[S,x] is Gödel’s constructible universe relativized to S and x. When using this definition, the ∞-Borel code ismade up of the set S and the formula φ, taken together.

29.4 References• W.H. Woodin The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal (1999 Walter deGruyter) p. 618

Chapter 30

Interval (mathematics)

This article is about intervals of real numbers and other totally ordered sets. For the most general definition, seepartially ordered set. For other uses, see Interval (disambiguation).

In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between twonumbers in the set is also included in the set. For example, the set of all numbers x satisfying 0 ≤ x ≤ 1 is an intervalwhich contains 0 and 1, as well as all numbers between them. Other examples of intervals are the set of all realnumbers R , the set of all negative real numbers, and the empty set.Real intervals play an important role in the theory of integration, because they are the simplest sets whose “size” or“measure” or “length” is easy to define. The concept of measure can then be extended to more complicated sets ofreal numbers, leading to the Borel measure and eventually to the Lebesgue measure.Intervals are central to interval arithmetic, a general numerical computing technique that automatically providesguaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical approximations,and arithmetic roundoff.Intervals are likewise defined on an arbitrary totally ordered set, such as integers or rational numbers. The notationof integer intervals is considered in the special section below.

30.1 Notations for intervals

The interval of numbers between a and b, including a and b, is often denoted [a, b]. The two numbers are called theendpoints of the interval. In countries where numbers are written with a decimal comma, a semicolon may be usedas a separator, to avoid ambiguity.

30.1.1 Including or excluding endpoints

To indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can be eitherreplaced with a parenthesis, or reversed. Both notations are described in International standard ISO 31-11. Thus, inset builder notation,

(a, b) = ]a, b[ = x ∈ R | a < x < b,[a, b) = [a, b[ = x ∈ R | a ≤ x < b,(a, b] = ]a, b] = x ∈ R | a < x ≤ b,[a, b] = [a, b] = x ∈ R | a ≤ x ≤ b.

Note that (a, a), [a, a), and (a, a] each represents the empty set, whereas [a, a] denotes the set a. When a > b, allfour notations are usually taken to represent the empty set.Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation(a, b) is often used to denote an ordered pair in set theory, the coordinates of a point or vector in analytic geometry

59

60 CHAPTER 30. INTERVAL (MATHEMATICS)

and linear algebra, or (sometimes) a complex number in algebra. That is why Bourbaki introduced the notation ]a,b[ to denote the open interval.[1] The notation [a, b] too is occasionally used for ordered pairs, especially in computerscience.Some authors use ]a, b[ to denote the complement of the interval (a, b); namely, the set of all real numbers that areeither less than or equal to a, or greater than or equal to b.

30.1.2 Infinite endpoints

In both styles of notation, one may use an infinite endpoint to indicate that there is no bound in that direction.Specifically, one may use a = −∞ or b = +∞ (or both). For example, (0, +∞) is the set of positive real numbers alsowritten ℝ+, and (−∞, +∞) is the set of real numbers ℝ.The extended real number line includes −∞ and +∞ as elements. The notations [−∞, b] , [−∞, b) , [a, +∞] , and (a,+∞] may be used in this context. For example (−∞, +∞] means the extended real numbers excluding only −∞.

30.1.3 Integer intervals

The notation [a .. b] when a and b are integers, or a .. b, or just a .. b is sometimes used to indicate the intervalof all integers between a and b, including both. This notation is used in some programming languages; in Pascal, forexample, it is used to formally define a subrange type, most frequently used to specify lower and upper bounds ofvalid indices of an array.An integer interval that has a finite lower or upper endpoint always includes that endpoint. Therefore, the exclusionof endpoints can be explicitly denoted by writing a .. b − 1 , a + 1 .. b , or a + 1 .. b − 1. Alternate-bracket notationslike [a .. b) or [a .. b[ are rarely used for integer intervals.

30.2 Terminology

An open interval does not include its endpoints, and is indicated with parentheses. For example (0,1) means greaterthan 0 and less than 1. A closed interval includes its endpoints, and is denoted with square brackets. For example[0,1] means greater than or equal to 0 and less than or equal to 1.A degenerate interval is any set consisting of a single real number. Some authors include the empty set in thisdefinition. A real interval that is neither empty nor degenerate is said to be proper, and has infinitely many elements.An interval is said to be left-bounded or right-bounded if there is some real number that is, respectively, smallerthan or larger than all its elements. An interval is said to be bounded if it is both left- and right-bounded; and is saidto be unbounded otherwise. Intervals that are bounded at only one end are said to be half-bounded. The emptyset is bounded, and the set of all reals is the only interval that is unbounded at both ends. Bounded intervals are alsocommonly known as finite intervals.Bounded intervals are bounded sets, in the sense that their diameter (which is equal to the absolute difference betweenthe endpoints) is finite. The diameter may be called the length, width,measure, or size of the interval. The size ofunbounded intervals is usually defined as +∞, and the size of the empty interval may be defined as 0 or left undefined.The centre (midpoint) of bounded interval with endpoints a and b is (a + b)/2, and its radius is the half-length |a −b|/2. These concepts are undefined for empty or unbounded intervals.An interval is said to be left-open if and only if it has no minimum (an element that is smaller than all other elements);right-open if it has no maximum; and open if it has both properties. The interval [0,1) = x | 0 ≤ x < 1, for example,is left-closed and right-open. The empty set and the set of all reals are open intervals, while the set of non-negativereals, for example, is a right-open but not left-open interval. The open intervals coincide with the open sets of thereal line in its standard topology.An interval is said to be left-closed if it has a minimum element, right-closed if it has a maximum, and simply closedif it has both. These definitions are usually extended to include the empty set and to the (left- or right-) unboundedintervals, so that the closed intervals coincide with closed sets in that topology.The interior of an interval I is the largest open interval that is contained in I; it is also the set of points in I which arenot endpoints of I. The closure of I is the smallest closed interval that contains I; which is also the set I augmented

30.3. CLASSIFICATION OF INTERVALS 61

with its finite endpoints.For any set X of real numbers, the interval enclosure or interval span of X is the unique interval that contains Xand does not properly contain any other interval that also contains X.

30.3 Classification of intervals

The intervals of real numbers can be classified into eleven different types, listed below; where a and b are real numbers,with a < b :

empty: [b, a] = (a, a) = [a, a) = (a, a] = = ∅degenerate: [a, a] = aproper and bounded:

(a, b) = x | a < x < b

[a, b] = x | a ≤ x ≤ b

[a, b) = x | a ≤ x < b

(a, b] = x | a < x ≤ b

left-bounded and right-unbounded:

(a,∞) = x |x > a

[a,∞) = x |x ≥ a

left-unbounded and right-bounded:

(−∞, b) = x |x < b

(−∞, b] = x |x ≤ b

unbounded at both ends: (−∞,+∞) = R

30.3.1 Intervals of the extended real line

In some contexts, an interval may be defined as a subset of the extended real numbers, the set of all real numbersaugmented with −∞ and +∞.In this interpretation, the notations [−∞, b] , [−∞, b) , [a, +∞] , and (a, +∞] are all meaningful and distinct. Inparticular, (−∞, +∞) denotes the set of all ordinary real numbers, while [−∞, +∞] denotes the extended reals.This choice affects some of the above definitions and terminology. For instance, the interval (−∞, +∞) = R is closedin the realm of ordinary reals, but not in the realm of the extended reals.

30.4 Properties of intervals

The intervals are precisely the connected subsets of R . It follows that the image of an interval by any continuousfunction is also an interval. This is one formulation of the intermediate value theorem.The intervals are also the convex subsets of R . The interval enclosure of a subset X ⊆ R is also the convex hull ofX .The intersection of any collection of intervals is always an interval. The union of two intervals is an interval if andonly if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other(e.g., (a, b) ∪ [b, c] = (a, c] ).If R is viewed as a metric space, its open balls are the open bounded sets (c + r, c − r), and its closed balls are theclosed bounded sets [c + r, c − r].

62 CHAPTER 30. INTERVAL (MATHEMATICS)

Any element x of an interval I defines a partition of I into three disjoint intervals I1, I2, I3: respectively, the elementsof I that are less than x, the singleton [x, x] = x , and the elements that are greater than x. The parts I1 and I3 areboth non-empty (and have non-empty interiors) if and only if x is in the interior of I. This is an interval version ofthe trichotomy principle.

30.5 Dyadic intervals

A dyadic interval is a bounded real interval whose endpoints are j2n and j+1

2n , where j and n are integers. Dependingon the context, either endpoint may or may not be included in the interval.Dyadic intervals have the following properties:

• The length of a dyadic interval is always an integer power of two.

• Each dyadic interval is contained in exactly one dyadic interval of twice the length.

• Each dyadic interval is spanned by two dyadic intervals of half the length.

• If two open dyadic intervals overlap, then one of them is a subset of the other.

The dyadic intervals consequently have a structure that reflects that of an infinite binary tree.Dyadic intervals are relevant to several areas of numerical analysis, including adaptive mesh refinement, multigridmethods and wavelet analysis. Another way to represent such a structure is p-adic analysis (for p = 2).[2]

30.6 Generalizations

30.6.1 Multi-dimensional intervals

In many contexts, an n -dimensional interval is defined as a subset ofRn that is the Cartesian product of n intervals,I = I1 × I2 × · · · × In , one on each coordinate axis.For n = 2 , this generally defines a rectangle whose sides are parallel to the coordinate axes; for n = 3 , it defines anaxis-aligned rectangular box.A facet of such an interval I is the result of replacing any non-degenerate interval factor Ik by a degenerate intervalconsisting of a finite endpoint of Ik . The faces of I comprise I itself and all faces of its facets. The corners of Iare the faces that consist of a single point of Rn .

30.6.2 Complex intervals

Intervals of complex numbers can be defined as regions of the complex plane, either rectangular or circular.[3]

30.7 Topological algebra

Intervals can be associated with points of the plane and hence regions of intervals can be associated with regions ofthe plane. Generally, an interval in mathematics corresponds to an ordered pair (x,y) taken from the direct product R× R of real numbers with itself. Often it is assumed that y > x. For purposes of mathematical structure, this restrictionis discarded,[4] and “reversed intervals” where y − x < 0 are allowed. Then the collection of all intervals [x,y] can beidentified with the topological ring formed by the direct sum of R with itself where addition and multiplication aredefined component-wise.The direct sum algebra (R ⊕ R,+,×) has two ideals, [x,0] : x ∈ R and [0,y] : y ∈ R . The identity elementof this algebra is the condensed interval [1,1]. If interval [x,y] is not in one of the ideals, then it has multiplicativeinverse [1/x, 1/y]. Endowed with the usual topology, the algebra of intervals forms a topological ring. The group ofunits of this ring consists of four quadrants determined by the axes, or ideals in this case. The identity component ofthis group is quadrant I.

30.8. SEE ALSO 63

Every interval can be considered a symmetric interval around its midpoint. In a reconfiguration published in 1956by M Warmus, the axis of “balanced intervals” [x, −x] is used along with the axis of intervals [x,x] that reduce to apoint. Instead of the direct sum R ⊕ R , the ring of intervals has been identified[5] with the split-complex numberplane by M. Warmus and D. H. Lehmer through the identification

z = (x + y)/2 + j (x − y)/2.

This linear mapping of the plane, which amounts of a ring isomorphism, provides the plane with a multiplicativestructure having some analogies to ordinary complex arithmetic, such as polar decomposition.

30.8 See also• Inequality

• Interval graph

• Interval finite element

30.9 References[1] http://hsm.stackexchange.com/a/193

[2] Kozyrev, Sergey (2002). “Wavelet theory as p-adic spectral analysis”. Izvestiya RAN. Ser. Mat. 66 (2): 149–158.doi:10.1070/IM2002v066n02ABEH000381. Retrieved 2012-04-05.

[3] Complex interval arithmetic and its applications, Miodrag Petković, Ljiljana Petković, Wiley-VCH, 1998, ISBN 978-3-527-40134-5

[4] Kaj Madsen (1979) Review of “Interval analysis in the extended interval space” by Edgar Kaucher from MathematicalReviews

[5] D. H. Lehmer (1956) Review of “Calculus of Approximations” from Mathematical Reviews

• T. Sunaga, “Theory of interval algebra and its application to numerical analysis”, In: Research Association ofApplied Geometry (RAAG) Memoirs, Ggujutsu Bunken Fukuy-kai. Tokyo, Japan, 1958, Vol. 2, pp. 29–46(547-564); reprinted in Japan Journal on Industrial and Applied Mathematics, 2009, Vol. 26, No. 2-3, pp.126–143.

30.10 External links• A Lucid Interval by Brian Hayes: An American Scientist article provides an introduction.

• Interval Notation Basics

• Interval computations website

• Interval computations research centers

• Interval Notation by George Beck, Wolfram Demonstrations Project.

• Weisstein, Eric W., “Interval”, MathWorld.

Chapter 31

Kleene–Brouwer order

In descriptive set theory, the Kleene–Brouwer order or Lusin–Sierpiński order[1] is a linear order on finite se-quences over some linearly ordered set (X,<) , that differs from the more commonly used lexicographic order inhow it handles the case when one sequence is a prefix of the other. In the Kleene–Brouwer order, the prefix is laterthan the longer sequence containing it, rather than earlier.The Kleene–Brouwer order generalizes the notion of a postorder traversal from finite trees to trees that are notnecessarily finite. For trees over a well-ordered set, the Kleene–Brouwer order is itself a well-ordering if and only ifthe tree has no infinite branch. It is named after Stephen Cole Kleene, Luitzen Egbertus Jan Brouwer, Nikolai Luzin,and Wacław Sierpiński.

31.1 Definition

If t and s are finite sequences of elements from X , we say that t <KB s when there is an n such that either:

• t n = s n and t(n) is defined but s(n) is undefined (i.e. t properly extends s ), or

• both s(n) and t(n) are defined, t(n) < s(n) , and t n = s n .

Here, the notation t n refers to the prefix of t up to but not including t(n) . In simple terms, t <KB s whenevers is a prefix of t (i.e. s terminates before t , and they are equal up to that point) or t is to the “left” of s on the firstplace they differ.[1]

31.2 Tree interpretation

A tree, in descriptive set theory, is defined as a set of finite sequences that is closed under prefix operations. Theparent in the tree of any sequence is the shorter sequence formed by removing its final element. Thus, any set of finitesequences can be augmented to form a tree, and the Kleene–Brouwer order is a natural ordering that may be givento this tree. It is a generalization to potentially-infinite trees of the postorder traversal of a finite tree: at every nodeof the tree, the child subtrees are given their left to right ordering, and the node itself comes after all its children.The fact that the Kleene–Brouwer order is a linear ordering (that is, that it is transitive as well as being total) followsimmediately from this, as any three sequences on which transitivity is to be tested form (with their prefixes) a finitetree on which the Kleene–Brouwer order coincides with the postorder.The significance of the Kleene–Brouwer ordering comes from the fact that if X is well-ordered, then a tree over Xis well-founded (having no infinitely long branches) if and only if the Kleene–Brouwer ordering is a well-ordering ofthe elements of the tree.[1]

64

31.3. RECURSION THEORY 65

31.3 Recursion theory

In recursion theory, the Kleene–Brouwer order may be applied to the computation trees of implementations of totalrecursive functionals. A computation tree is well-founded if and only if the computation performed by it is totalrecursive. Each state x in a computation tree may be assigned an ordinal number ||x|| , the supremum of the ordi-nal numbers 1 + ||y|| where y ranges over the children of x in the tree. In this way, the total recursive functionalsthemselves can be classified into a hierarchy, according to the minimum value of the ordinal at the root of a com-putation tree, minimized over all computation trees that implement the functional. The Kleene–Brouwer order of awell-founded computation tree is itself a recursive well-ordering, and at least as large as the ordinal assigned to thetree, from which it follows that the levels of this hierarchy are indexed by recursive ordinals.[2]

31.4 History

This ordering was used by Lusin & Sierpinski (1923),[3] and then again by Brouwer (1924).[4] Brouwer does notcite any references, but Moschovakis argues that he may either have seen Lusin & Sierpinski (1923), or have beeninfluenced by earlier work of the same authors leading to this work. Much later, Kleene (1955) studied the sameordering, and credited it to Brouwer.[5]

31.5 References[1] Moschovakis, Yiannis (2009), Descriptive Set Theory (2nd ed.), Rhode Island: American Mathematical Society, pp. 148–

149, 203–204, ISBN 978-0-8218-4813-5

[2] Schwichtenberg, Helmut; Wainer, Stanley S. (2012), “2.8 Recursive type-2 functionals and well-foundedness”, Proofs andcomputations, Perspectives in Logic, Cambridge: Cambridge University Press, pp. 98–101, ISBN 978-0-521-51769-0,MR 2893891.

[3] Lusin, Nicolas; Sierpinski, Waclaw (1923), “Sur un ensemble non measurable B”, Journal de Mathématiques Pure et Ap-pliquées 9 (2): 53–72.

[4] Brouwer, L. E. J. (1924), “Beweis, dass jede volle Funktion gleichmässig stetig ist”, Koninklijke Nederlandse Akademie vanWetenschappen, Proc. Section of Sciences 27: 189–193. As cited by Kleene (1955).

[5] Kleene, S. C. (1955), “On the forms of the predicates in the theory of constructive ordinals. II”, American Journal of Math-ematics 77: 405–428, doi:10.2307/2372632, JSTOR 2372632, MR 0070595. See in particular section 26, “A digressionconcerning recursive linear orderings”, pp. 419–422.

Chapter 32

Kuratowski–Ulam theorem

In mathematics, theKuratowski–Ulam theorem, introduced by Kazimierz Kuratowski and Stanislaw Ulam (1932),called also Fubini theorem for category, is an analog of the Fubini’s theorem for arbitrary second countable Bairespaces. Let X and Y be second countable Baire spaces (or, in particular, Polish spaces), and A ⊂ X × Y . Then thefollowing are equivalent if A has the Baire property:

1. A is meager (respectively comeager)

2. The set x ∈ X : Ax in comeager) (resp. meager is Y is comeager in X, where Ax = πY [A ∩ x × Y ] ,where πY is the projection onto Y.

Even if A does not have the Baire property, 2. follows from 1.[1] Note that the theorem still holds (perhaps vacuously)for X - arbitrary Hausdorff space and Y - Hausdorff with countable π-base.The theorem is analogous to regular Fubini’s theorem for the case where the considered function is a characteristicfunction of a set in a product space, with usual correspondences – meagre set with set of measure zero, comeagre setwith one of full measure, a set with Baire property with a measurable set.

32.1 References[1] Srivastava, S. (1998). A Course on Borel Sets. Berlin: Springer. p. 112. ISBN 0-387-98412-7.

• Kuratowski, C.; Ulam, St. (1932), “Quelques propriétés topologiques du produit combinatoire” (PDF), Fun-damenta Mathematicae (Institute of Mathematics Polish Academy of Sciences) 19 (1): 247–251

66

Chapter 33

L(R)

In set theory, L(R) (pronounced L of R) is the smallest transitive inner model of ZF containing all the ordinals andall the reals.

33.1 Construction

It can be constructed in a manner analogous to the construction of L (that is, Gödel’s constructible universe), byadding in all the reals at the start, and then iterating the definable powerset operation through all the ordinals.

33.2 Assumptions

In general, the study of L(R) assumes a wide array of large cardinal axioms, since without these axioms one cannotshow even that L(R) is distinct from L. But given that sufficient large cardinals exist, L(R) does not satisfy the axiomof choice, but rather the axiom of determinacy. However, L(R) will still satisfy the axiom of dependent choice, givenonly that the von Neumann universe, V, also satisfies that axiom.

33.3 Results

Some additional results of the theory are:

• Every projective set of reals -- and therefore every analytic set and every Borel set of reals -- is an element ofL(R).

• Every set of reals in L(R) is Lebesgue measurable (in fact, universally measurable) and has the property ofBaire and the perfect set property.

• L(R) does not satisfy the axiom of uniformization or the axiom of real determinacy.

• R#, the sharp of the set of all reals, has the smallest Wadge degree of any set of reals not contained in L(R).

• While not every relation on the reals in L(R) has a uniformization in L(R), every such relation does have auniformization in L(R#).

• Given any (set-size) generic extension V[G] of V, L(R) is an elementary submodel of L(R) as calculated inV[G]. Thus the theory of L(R) cannot be changed by forcing.

• L(R) satisfies AD+.

67

68 CHAPTER 33. L(R)

33.4 References• Woodin, W. Hugh (1988). “Supercompact cardinals, sets of reals, and weakly homogeneous trees”. Proceed-

ings of the National Academy of Sciences of theUnited States of America 85 (18): 6587–6591. doi:10.1073/pnas.85.18.6587.PMC 282022. PMID 16593979.

Chapter 34

Lightface analytic game

In descriptive set theory, a lightface analytic game is a game whose payoff set A is a Σ11 subset of Baire space; that

is, there is a tree T on ω × ω which is a computable subset of (ω × ω)<ω , such that A is the projection of the set ofall branches of T.The determinacy of all lightface analytic games is equivalent to the existence of 0#.

69

Chapter 35

List of properties of sets of reals

This page lists some properties of sets of real numbers. The general study of these concepts forms descriptive settheory, which has a rather different emphasis from general topology.

35.1 Definability properties

• Borel set

• Analytic set

• C-measurable set

• Projective set

• Inductive set

• Infinity-Borel set

• Suslin set

• Homogeneously Suslin set

• Weakly homogeneously Suslin set

• Set of uniqueness

35.2 Regularity properties

• Property of Baire

• Lebesgue measurable

• Universally measurable set

• Perfect set property

• Universally Baire set

35.3 Largeness and smallness properties

• Meager set

• Comeager set - A comeager set is one whose complement is meager.

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35.3. LARGENESS AND SMALLNESS PROPERTIES 71

• Null set

• Conull set

• Dense set

• Nowhere dense set

Chapter 36

Lusin’s separation theorem

This article is about the separation theorem. For the theorem on continuous functions, see Lusin’s theorem.

In descriptive set theory and mathematical logic, Lusin’s separation theorem states that if A and B are disjointanalytic subsets of Polish space, then there is a Borel set C in the space such that A ⊆ C and B ∩ C = ∅.[1] It is namedafter Nikolai Luzin, who proved it in 1927.[2]

The theorem can be generalized to show that for each sequence (An) of disjoint analytic sets there is a sequence (Bn)of disjoint Borel sets such that An ⊆ Bn for each n. [1]

An immediate consequence is Suslin’s theorem, which states that if a set and its complement are both analytic, thenthe set is Borel.

36.1 Notes[1] (Kechris 1995, p. 87).

[2] (Lusin 1927).

36.2 References• Kechris, Alexander (1995), Classical descriptive set theory, Graduate texts inmathematics 156, Berlin–Heidelberg–New York: Springer–Verlag, pp. xviii+402, doi:10.1007/978-1-4612-4190-4, ISBN 0-387-94374-9, MR1321597, Zbl 0819.04002 (ISBN 3-540-94374-9 for the European edition)

• Lusin, Nicolas (1927), “Sur les ensembles analytiques” (PDF), Fundamenta Mathematicae (in French) 10:1–95, JFM 53.0171.05.

72

Chapter 37

Luzin space

For continuous images of separable complete metric spaces, known as Lusin spaces, see Polish space #Lusin spaces.

In mathematics, a Luzin space (or Lusin space), named for N. N. Luzin, is an uncountable topological T1 spacewithout isolated points in which every nowhere-dense subset is countable. There are many minor variations of thisdefinition in use: the T1 condition can be replaced by T2 or T3, and some authors allow a countable or even arbitrarynumber of isolated points.The existence of a Luzin space is independent of the axioms of ZFC. Luzin (1914) showed that the continuumhypothesis implies that a Luzin space exists. Kunen (1977) showed that assuming Martin’s Axiom and the negationof the continuum hypothesis, there are no Hausdorff Luzin spaces.

37.1 In real analysis

In real analysis and descriptive set theory, a Luzin set (or Lusin set), is defined as an uncountable subset A of thereals such that every uncountable subset of A is nonmeager; that is, of second Baire category. Equivalently, A is anuncountable set of reals which meets every first category set in only countably many points. Luzin proved that, ifthe continuum hypothesis holds, then every nonmeager set has a Luzin subset. Obvious properties of a Luzin set arethat it must be nonmeager (otherwise the set itself is an uncountable meager subset) and of measure zero, becauseevery set of positive measure contains a meager set which also has positive measure, and is therefore uncountable.A weakly Luzin set is an uncountable subset of a real vector space such that for any uncountable subset the set ofdirections between different elements of the subset is dense in the sphere of directions.The measure-category duality provides a measure analogue of Luzin sets – sets of positive outer measure, everyuncountable subset of which has positive outer measure. These sets are called Sierpiński sets, afterWacław Sierpiński.Sierpiński sets are weakly Luzin sets but are not Luzin sets.

37.2 Example of a Luzin set

Choose a collection of 2ℵ0 meager subsets of R such that every meager subset is contained in one of them. By thecontinuum hypothesis, it is possible to enumerate them as Sα for countable ordinals α. For each countable ordinal βchoose a real number xᵦ that is not in any of the sets Sα for α<β, which is possible as the union of these sets is meagerso is not the whole of R. Then the uncountable set X of all these real numbers xᵦ has only a countable number ofelements in each set Sα, so is a Luzin set.More complicated variations of this construction produce examples of Luzin sets that are subgroups, subfields orreal-closed subfields of the real numbers.

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74 CHAPTER 37. LUZIN SPACE

37.3 References• Arkhangelskii, A V (1978), “STRUCTURE AND CLASSIFICATION OF TOPOLOGICAL SPACES ANDCARDINAL INVARIANTS”,RussianMathematical Surveys 33 (6): 33–96, doi:10.1070/RM1978v033n06ABEH003884Paper mentioning Luzin spaces

• Efimov, B.A. (2001), “Luzin space”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• Kunen, Kenneth (1977), “Luzin spaces”, Topology Proceedings, Vol. I (Conf., Auburn Univ., Auburn, Ala.,1976), pp. 191–199, MR 0450063

• Lusin, N.N. (1914), “Sur un problème de M. Baire”, C.R. Acad. Sci. Paris 158: 1258–1261

• Oxtoby, John C. (1980), Measure and category: a survey of the analogies between topological and measurespaces, Berlin: Springer-Verlag, ISBN 0-387-90508-1

Chapter 38

Martin measure

In descriptive set theory, the Martin measure is a filter on the set of Turing degrees of sets of natural numbers,named after Donald A. Martin. Under the axiom of determinacy it can be shown to be an ultrafilter.

38.1 Definition

Let D be the set of Turing degrees of sets of natural numbers. Given some equivalence class [X] ∈ D , we maydefine the cone (or upward cone) of [X] as the set of all Turing degrees [Y ] such that X ≤T Y ; that is, the set ofTuring degrees which are “more complex” than X under Turing reduction.We say that a set A of Turing degrees has measure 1 under the Martin measure exactly when A contains some cone.Since it is possible, for any A , to construct a game in which player I has a winning strategy exactly when A containsa cone and in which player II has a winning strategy exactly when the complement of A contains a cone, the axiomof determinacy implies that the measure-1 sets of Turing degrees form an ultrafilter.

38.2 Consequences

It is easy to show that a countable intersection of cones is itself a cone; the Martin measure is therefore a countablycomplete filter. This fact, combined with the fact that the Martin measure may be transferred to ω1 by a simplemapping, tells us that ω1 is measurable under the axiom of determinacy. This result shows part of the importantconnection between determinacy and large cardinals.

38.3 References• Moschovakis, Yiannis N. (2009). Descriptive Set Theory. Mathematical surveys and monographs 155 (2nded.). American Mathematical Society. p. 338. ISBN 9780821848135.

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Chapter 39

Meagre set

In the mathematical fields of general topology and descriptive set theory, ameagre set (also called ameager set or aset of first category) is a set that, considered as a subset of a (usually larger) topological space, is in a precise sensesmall or negligible. The meagre subsets of a fixed space form a sigma-ideal of subsets; that is, any subset of a meagreset is meagre, and the union of countably many meagre sets is meagre.General topologists use the term Baire space to refer to a broad class of topological spaces on which the notion ofmeagre set is not trivial (in particular, the entire space is not meagre). Descriptive set theorists mostly study meagresets as subsets of the real numbers, or more generally any Polish space, and reserve the term Baire space for oneparticular Polish space.The complement of a meagre set is a comeagre set or residual set.

39.1 Definition

Given a topological space X, a subset A of X is meagre if it can be expressed as the union of countably many nowheredense subsets of X. Dually, a comeagre set is one whose complement is meagre, or equivalently, the intersection ofcountably many sets with dense interiors.A subset B of X is nowhere dense if there is no neighbourhood on which B is dense: for any nonempty open set Uin X, there is a nonempty open set V contained in U such that V and B are disjoint.The complement of a nowhere dense set is a dense set. More precisely, the complement of a nowhere dense set is aset with dense interior. Not every dense set has a nowhere dense complement. The complement of a dense set canhave nowhere dense, and dense regions.

39.1.1 Relation to Borel hierarchy

Just as a nowhere dense subset need not be closed, but is always contained in a closed nowhere dense subset (viz, itsclosure), a meagre set need not be an Fσ set (countable union of closed sets), but is always contained in an Fσ setmade from nowhere dense sets (by taking the closure of each set).Dually, just as the complement of a nowhere dense set need not be open, but has a dense interior (contains a denseopen set), a comeagre set need not be a Gδ set (countable intersection of open sets), but contains a dense Gδ setformed from dense open sets.

39.2 Terminology

A meagre set is also called a set of first category; a nonmeagre set (that is, a set that is not meagre) is also called a setof second category. Second category does not mean comeagre – a set may be neither meagre nor comeagre (in thiscase it will be of second category).

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39.3. PROPERTIES 77

39.3 Properties

• Any subset of a meagre set is meagre; any superset of a comeagre set is comeagre.

• The union of countably many meagre sets is also meagre; the intersection of countably many comeagre sets iscomeagre.

This follows from the fact that a countable union of countable sets is countable.

• Banach Category Theorem: In any space X, the union of any family of open sets of the first category is of thefirst category.[1]

39.4 Banach–Mazur game

Meagre sets have a useful alternative characterization in terms of the Banach–Mazur game. If Y is a topologicalspace,W is a family of subsets of Y which have nonempty interior such that every nonempty open set has a subset inW , andX is any subset of Y , then there is a Banach-Mazur game corresponding toX,Y,W . In the Banach-Mazurgame, two players, P1 and P2 , alternate choosing successively smaller (in terms of the subset relation) elements ofW to produce a descending sequenceW1 ⊃W2 ⊃W3 ⊃ · · · . If the intersection of this sequence contains a pointin X , P1 wins; otherwise, P2 wins. If W is any family of sets meeting the above criteria, then P2 has a winningstrategy if and only if X is meagre.

39.5 Examples

39.5.1 Subsets of the reals

• The rational numbers are meagre as a subset of the reals and as a space – that is, they do not form a Bairespace.

• The Cantor set is meagre as a subset of the reals, but not as a space, since it is a complete metric space and isthus a Baire space, by the Baire category theorem.

39.5.2 Function spaces

• The set of functions which have a derivative at some point is a meagre set in the space of all continuousfunctions.[2]

39.6 See also

• Baire category theorem

• Generic property, for analogs to residual

• Negligible set, for analogs to meagre

39.7 Notes[1] Oxtoby, John C. (1980). “The Banach Category Theorem”. Measure and Category (Second ed.). New York: Springer. pp.

62–65. ISBN 0-387-90508-1.

[2] Banach, S. (1931). "Über die Baire’sche Kategorie gewisser Funktionenmengen”. Studia. Math. 3 (1): 174–179.

78 CHAPTER 39. MEAGRE SET

39.8 External links• Is there a measure zero set which isn’t meagre?

Chapter 40

Negative-dimensional space

In topology, a discipline within mathematics, a negative-dimensional space is an extension of the usual notion ofspace allowing for negative dimensions.[1]

40.1 Definition

Suppose that Mt0 is a compact space of Hausdorff dimension t0, which is an element of a scale of compact spacesembedded in each other and parametrized by t (0 < t < ∞). Such scales are considered equivalent with respect toMt0if the compact spaces constituting them coincide for t ≥ t0. It is said that the compact space Mt0 is the hole in thisequivalent set of scales, and −t0 is the negative dimension of the corresponding equivalence class.[2]

40.2 History

By the 1940s, the science of topology had developed and studied a thorough basic theory of topological spaces ofpositive dimension. Motivated by computations, and to some extent aesthetics, topologists searched for mathematicalframeworks that extended our notion of space to allow for negative dimensions. Such dimensions, as well as thefourth and higher dimensions, are hard to imagine since we are not able to directly observe them. It wasn’t until the1960s that a special topological framework was constructed—the category of spectra. A spectrum is a generalizationof space that allows for negative dimensions. The concept of negative-dimensional spaces is applied, for example, toanalyze linguistic statistics.[3]

40.3 See also

• Cone (topology)

• Equidimensionality

• Join (topology)

• Suspension/desuspension

• Spectrum (topology)

40.4 References[1] Wolcott, Luke; McTernan, Elizabeth (2012). “ImaginingNegative-Dimensional Space” (PDF). In Bosch, Robert; McKenna,

Douglas; Sarhangi, Reza. Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture. Phoenix, Arizona,USA: Tessellations Publishing. pp. 637–642. ISBN 978-1-938664-00-7. ISSN 1099-6702. Retrieved 25 June 2015.

79

80 CHAPTER 40. NEGATIVE-DIMENSIONAL SPACE

[2] Maslov, V.P. “General Notion of a Topological Space ofNegativeDimension andQuantization of Its Density”. springer.com.Retrieved 2015-06-23.

[3] Maslov, V.P. “Negative Dimension in General and Asymptotic Topology”. arxiv.org. Retrieved 2015-06-25.

40.5 External links• Отрицательная асимптотическая топологическая размерность, новый конденсат и их связь с квантованнымзаконом Ципфа. For a translation into English, see Maslov, V.P. (November 2006). “Negative asymptotictopological dimension, a new condensate, and their relation to the quantized Zipf law”. Mathematical Notes 80(5-6): 806–813. Retrieved 30 June 2015.

Chapter 41

Nested intervals

0

0

0

0

In mathematics, a sequence of nested intervals is understood as a collection of sets of real numbers

In

such that each set In is an interval of the real line, for n = 1, 2, 3, ..., and that further

In ₊ ₁ is a subset of In

for all n. In other words, the intervals diminish, with the left-hand end moving only towards the right, and the right-hand end only to the left.The main question to be posed is the nature of the intersection of all the In. Without any further information, all thatcan be said is that the intersection J of all the In, i.e. the set of all points common to the intervals, is either the emptyset, a point, or some interval.The possibility of an empty intersection can be illustrated by the intersection when In is the open interval

(0, 2−n).

Here the intersection is empty, because no number x is both greater than 0 and less than every fraction 2−n.The situation is different for closed intervals. The nested intervals theorem states that if each In is a closed and boundedinterval, say

In = [an, bn]

with

81

82 CHAPTER 41. NESTED INTERVALS

an ≤ bn

then under the assumption of nesting, the intersection of the In is not empty. It may be a singleton set c, or anotherclosed interval [a, b]. More explicitly, the requirement of nesting means that

an ≤ an ₊ ₁

and

bn ≥ bn ₊ ₁.

Moreover, if the length of the intervals converges to 0, then the intersection of the In is a singleton.One can consider the complement of each interval, written as (−∞, an) ∪ (bn,∞) . By De Morgan’s laws, thecomplement of the intersection is a union of two disjoint open sets. By the connectedness of the real line there mustbe something between them. This shows that the intersection of (even an uncountable number of) nested, closed, andbounded intervals is nonempty.

41.1 Higher dimensions

In two dimensions there is a similar result: nested closed disks in the plane must have a common intersection. Thisresult was shown by Hermann Weyl to classify the singular behaviour of certain differential equations.

41.2 See also• Bisection

• Cantor’s Intersection Theorem

41.3 References• Fridy, J. A. (2000), “3.3 The Nested Intervals Theorem”, Introductory Analysis: The Theory of Calculus,Academic Press, p. 29, ISBN 9780122676550.

• Shilov, Georgi E. (2012), “1.8 The Principle of Nested Intervals”, Elementary Real and Complex Analysis,Dover Books on Mathematics, Courier Dover Publications, pp. 21–22, ISBN 9780486135007.

• Sohrab, Houshang H. (2003), “Theorem 2.1.5 (Nested Intervals Theorem)", Basic Real Analysis, Springer, p.45, ISBN 9780817642112.

Chapter 42

Normal number

For the floating-point meaning in computing, see normal number (computing).

In mathematics, a normal number is a real number whose infinite sequence of digits in every base b[1] is distributeduniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b2 pairs ofdigits are equally likely with density b−2, all b3 triplets of digits equally likely with density b−3, etc.Intuitively this means that no digit, or combination of digits, occurs more frequently than any other, and this is truewhether the number is written in base 10, binary, or any other base. A normal number can be thought of as an infinitesequence of coin flips (binary) or rolls of a die (base 6). Even though therewill be sequences such as 10, 100, or moreconsecutive tails (binary) or fives (base 6) or even 10, 100, or more repetitions of a sequence such as tail-head (twoconsecutive coin flips) or 6-1 (two consecutive rolls of a die), there will also be equally many of any other sequenceof equal length. No digit or sequence is “favored”.While a general proof can be given that almost all real numbers are normal (in the sense that the set of exceptionshas Lebesgue measure zero), this proof is not constructive and only very few specific numbers have been shown tobe normal. For example, it is widely believed that the numbers √2, π, and e are normal, but a proof remains elusive.

42.1 Definitions

Let Σ be a finite alphabet of b digits, and Σ∞ the set of all sequences that may be drawn from that alphabet. Let S∈ Σ∞ be such a sequence. For each a in Σ let NS(a, n) denote the number of times the letter a appears in the first ndigits of the sequence S. We say that S is simply normal if the limit

limn→∞

NS(a, n)

n=

1

b

for each a. Now let w be any finite string in Σ∗ and let NS(w, n) to be the number of times the string w appears as asubstring in the first n digits of the sequence S. (For instance, if S = 01010101..., then NS(010, 8) = 3.) S is normalif, for all finite strings w ∈ Σ∗,

limn→∞

NS(w, n)

n=

1

b|w|

where | w | denotes the length of the string w. In other words, S is normal if all strings of equal length occur withequal asymptotic frequency. For example, in a normal binary sequence (a sequence over the alphabet 0,1), 0 and 1each occur with frequency 1⁄2; 00, 01, 10, and 11 each occur with frequency 1⁄4; 000, 001, 010, 011, 100, 101, 110,and 111 each occur with frequency 1⁄8, etc. Roughly speaking, the probability of finding the string w in any givenposition in S is precisely that expected if the sequence had been produced at random.Suppose now that b is an integer greater than 1 and x is a real number. Consider the infinite digit sequence expansionSx, b of x in the base b positional number system (we ignore the decimal point). We say that x is simply normal in

83

84 CHAPTER 42. NORMAL NUMBER

base b if the sequence Sx, b is simply normal[2] and that x is normal in base b if the sequence Sx, b is normal.[3]The number x is called a normal number (or sometimes an absolutely normal number) if it is normal in base bfor every integer b greater than 1.[4][5]

A given infinite sequence is either normal or not normal, whereas a real number, having a different base-b expansionfor each integer b ≥ 2, may be normal in one base but not in another.[6][7] For bases r and s with log r / log s rational(so that r = bm and s = bn) every number normal in base r is normal in base s. For bases r and s with log r / log sirrational, there are uncountably many numbers normal in each base but not the other.[7]

A disjunctive sequence is a sequence in which every finite string appears. A normal sequence is disjunctive, but adisjunctive sequence need not be normal. A rich number in base b is one whose expansion in base b is disjunctive:[8]one that is disjunctive to every base is called absolutely disjunctive or is said to be a lexicon. A number normal in baseb is rich in base b, but not necessarily conversely. The real number x is rich in base b if and only if the set x bn mod1: n∈N is dense in the unit interval.[8][9]

We defined a number to be simply normal in base b if each individual digit appears with frequency 1/b. For a givenbase b, a number can be simply normal (but not normal or b-dense), b-dense (but not simply normal or normal),normal (and thus simply normal and b-dense), or none of these. A number is absolutely non-normal or absolutelyabnormal if it is not simply normal in any base.[4][10]

42.2 Properties and examples

The concept of a normal number was introduced by Émile Borel in 1909. Using the Borel–Cantelli lemma, he provedthe normal number theorem: almost all real numbers are normal, in the sense that the set of non-normal numbers hasLebesgue measure zero (Borel 1909). This theorem established the existence of normal numbers. In 1917, WacławSierpiński showed that it is possible to specify a particular such number. Becher and Figueira proved in 2002 thatthere is a computable absolutely normal number, however no digits of their number are known.The set of non-normal numbers, though “small” in the sense of being a null set, is “large” in the sense of beinguncountable. For instance, there are uncountably many numbers whose decimal expansion does not contain the digit5, and none of these are normal.Champernowne’s number

0.1234567891011121314151617...,

obtained by concatenating the decimal representations of the natural numbers in order, is normal in base 10, but itmight not be normal in some other bases.The Copeland–Erdős constant

0.235711131719232931374143...,

obtained by concatenating the prime numbers in base 10, is normal in base 10, as proved by Copeland and Erdős(1946). More generally, the latter authors proved that the real number represented in base b by the concatenation

0.f(1)f(2)f(3)...,

where f(n) is the nth prime expressed in base b, is normal in base b. Besicovitch (1935) proved that the numberrepresented by the same expression, with f(n) = n2,

0.149162536496481100121144...,

obtained by concatenating the square numbers in base 10, is normal in base 10. Davenport & Erdős (1952) provedthat the number represented by the same expression, with f being any polynomial whose values on the positive integersare positive integers, expressed in base 10, is normal in base 10.Nakai & Shiokawa (1992) proved that if f(x) is any non-constant polynomial with real coefficients such that f(x) > 0for all x > 0, then the real number represented by the concatenation

42.2. PROPERTIES AND EXAMPLES 85

0.[f(1)][f(2)][f(3)]...,

where [f(n)] is the integer part of f(n) expressed in base b, is normal in base b. (This result includes as special casesall of the above-mentioned results of Champernowne, Besicovitch, and Davenport & Erdős.) The authors also showthat the same result holds even more generally when f is any function of the form

f(x) = α·xβ + α1·xβ1 + ... + αd·xβd,

where the αs and βs are real numbers with β > β1 > β2 > ... > β ≥ 0, and f(x) > 0 for all x > 0.Every Chaitin’s constant Ω is a normal number (Calude, 1994). A computable normal number was constructed in(Becher 2002). Although these constructions do not directly give the digits of the numbers constructed, the secondshows that it is possible in principle to enumerate all the digits of a particular normal number.Bailey and Crandall show an explicit uncountably infinite class of b-normal numbers by perturbing Stoneham num-bers.[11]

It has been an elusive goal to prove the normality of numbers which were not explicitly constructed for the purpose.It is for instance unknown whether √2, π, ln(2) or e is normal (but all of them are strongly conjectured to be normal,because of some empirical evidence ). It is not even known whether all digits occur infinitely often in the decimalexpansions of those constants. In particular, the popular claim “every string of numbers eventually occurs in π" is notknown to be true. It has been conjectured that every irrational algebraic number is normal; while no counterexamplesare known, there also exists no algebraic number that has been proven to be normal in any base.

42.2.1 Non-normal numbers

No rational number is normal to any base, since the digit sequences of rational numbers are eventually periodic.[12](However, a rational number can be simply normal to a particular base: 123,456,789

9,999,999,999 = 0.0123456789 is simplynormal to base 10.)Martin 2001 has given a simple example of an irrational absolutely non-normal number.[13] Let d2 = 4 and

dj = jdj−1/(j−1) ,

ξ =

∞∏j=2

(1− 1

dj

).

Then ξ is absolutely non-normal and a Liouville number; hence a transcendental number.

42.2.2 Properties

Additional properties of normal numbers include:

• Every positive number x is the product of two normal numbers. For instance if y is chosen uniformly at randomfrom the interval (0,1) then almost surely y and x/y are both normal, and their product is x.

• If x is normal in base b and q ≠ 0 is a rational number, then x · q is normal in base b. (Wall 1949)

• If A ⊆ N is dense (for every α < 1 and for all sufficiently large n, |A∩ 1, . . . , n| ≥ nα ) and a1, a2, a3, . . .are the base-b expansions of the elements of A, then the number 0.a1a2a3 . . . , formed by concatenating theelements of A, is normal in base b (Copeland and Erdős 1946). From this it follows that Champernowne’snumber is normal in base 10 (since the set of all positive integers is obviously dense) and that the Copeland–Erdős constant is normal in base 10 (since the prime number theorem implies that the set of primes is dense).

• A sequence is normal if and only if every block of equal length appears with equal frequency. (A block oflength k is a substring of length k appearing at a position in the sequence that is a multiple of k: e.g. the firstlength-k block in S is S[1..k], the second length-k block is S[k+1..2k], etc.) This was implicit in the work ofZiv and Lempel (1978) and made explicit in the work of Bourke, Hitchcock, and Vinodchandran (2005).

86 CHAPTER 42. NORMAL NUMBER

• A number is normal in base b if and only if it is simply normal in base bk for every integer k ≥ 1 . This followsfrom the previous block characterization of normality: Since the nth block of length k in its base b expansioncorresponds to the nth digit in its base bk expansion, a number is simply normal in base bk if and only if blocksof length k appear in its base b expansion with equal frequency.

• A number is normal if and only if it is simply normal in every base. This follows from the previous character-ization of base b normality.

• A number is b-normal if and only if there exists a set of positive integers m1 < m2 < m3 < · · · where thenumber is simply normal to bases bm for all m ∈ m1,m2, . . .. [14] No finite set suffices to show that thenumber is b-normal.

• The set of normal sequences is closed under finite variations: adding, removing, or changing a finite numberof digits in any normal sequence leaves it normal.

42.3 Connection to finite-state machines

Agafonov showed an early connection between finite-statemachines and normal sequences: every infinite subsequenceselected from a normal sequence by a regular language is also normal. In other words, if one runs a finite-state machineon a normal sequence, where each of the finite-state machine’s states are labeled either “output” or “no output”, andthe machine outputs the digit it reads next after entering an “output” state, but does not output the next digit afterentering a “no output state”, then the sequence it outputs will be normal (Agafonov 1968).A deeper connection exists with finite-state gamblers (FSGs) and information lossless finite-state compressors (ILF-SCs).

• A finite-state gambler (a.k.a. finite-state martingale) is a finite-state machine over a finite alphabetΣ , eachof whose states is labelled with percentages of money to bet on each digit in Σ . For instance, for an FSG overthe binary alphabet Σ = 0, 1 , the current state q bets some percentage q0 ∈ [0, 1] of the gambler’s moneyon the bit 0, and the remaining q1 = 1 − q0 fraction of the gambler’s money on the bit 1. The money beton the digit that comes next in the input (total money times percent bet) is multiplied by |Σ| , and the rest ofthe money is lost. After the bit is read, the FSG transitions to the next state according to the input it received.A FSG d succeeds on an infinite sequence S if, starting from $1, it makes unbounded money betting on thesequence; i.e., if

lim supn→∞

d(S n) = ∞,

where d(S n) is the amount of money the gambler d has after reading the first n digits of S (see limitsuperior).

• A finite-state compressor is a finite-state machine with output strings labelling its state transitions, includingpossibly the empty string. (Since one digit is read from the input sequence for each state transition, it isnecessary to be able to output the empty string in order to achieve any compression at all). An informationlossless finite-state compressor is a finite-state compressor whose input can be uniquely recovered from itsoutput and final state. In other words, for a finite-state compressor C with state set Q, C is information losslessif the function f : Σ∗ → Σ∗ × Q , mapping the input string of C to the output string and final state of C,is 1–1. Compression techniques such as Huffman coding or Shannon–Fano coding can be implemented withILFSCs. An ILFSC C compresses an infinite sequence S if

lim infn→∞

|C(S n)|n

< 1,

42.4. CONNECTION TO EQUIDISTRIBUTED SEQUENCES 87

where |C(S n)| is the number of digits output by C after reading the first n digits of S. Note that thecompression ratio (the limit inferior above) can always be made to equal 1 by the 1-state ILFSC thatsimply copies its input to the output.

Schnorr and Stimm showed that no FSG can succeed on any normal sequence, and Bourke, Hitchcock and Vinod-chandran showed the converse. Therefore:

A sequence is normal if and only if there is no finite-state gambler that succeeds on it.

Ziv and Lempel showed:

A sequence is normal if and only if it is incompressible by any information lossless finite-state compressor

(they actually showed that the sequence’s optimal compression ratio over all ILFSCs is exactly its entropy rate, aquantitative measure of its deviation from normality, which is 1 exactly when the sequence is normal). Since theLZ compression algorithm compresses asymptotically as well as any ILFSC, this means that the LZ compressionalgorithm can compress any non-normal sequence. (Ziv Lempel 1978)These characterizations of normal sequences can be interpreted to mean that “normal” = “finite-state random"; i.e.,the normal sequences are precisely those that appear random to any finite-state machine. Compare this with thealgorithmically random sequences, which are those infinite sequences that appear random to any algorithm (and infact have similar gambling and compression characterizations with Turing machines replacing finite-state machines).

42.4 Connection to equidistributed sequences

A number x is normal in base b if and only if the sequence(bkx

)∞k=0

is equidistributed modulo 1,[15][16] or equiva-lently, using Weyl’s criterion, if and only if

limn→∞

1

n

n−1∑k=0

e2πimbkx = 0 integers all form ≥ 1.

This connection leads to the terminology that x is normal in base β for any real number β if the sequence(xβk

)∞k=0

is equidistributed modulo 1.[16]

42.5 Notes[1] The only bases considered here are natural numbers greater than 1

[2] Bugeaud 2012, p. 78

[3] Bugeaud 2012, p. 79

[4] Bugeaud 2012, p. 102

[5] Adamczewski & Bugeaud 2010, p. 413

[6] Cassels 1959

[7] Schmidt 1960

[8] Bugeaud 2012, p. 92

[9] x bn mod 1 denotes the fractional part of x bn.

[10] Martin (2001)

[11] Bailey & Crandall (2002).

88 CHAPTER 42. NORMAL NUMBER

[12] Murty (2007, p. 483).

[13] Bugeaud (2012) p.113

[14] Long (1957).

[15] Bugeaud 2012, p. 89

[16] Everest et al. 2003, p. 127

42.6 See also• Champernowne constant

• De Bruijn sequence

• Infinite monkey theorem

• The Library of Babel

• Illegal number

42.7 References• Adamczewski, Boris; Bugeaud, Yann (2010), “8. Transcendence and diophantine approximation”, in Berthé,Valérie; Rigo, Michael, Combinatorics, automata, and number theory, Encyclopedia of Mathematics and itsApplications 135, Cambridge: Cambridge University Press, pp. 410–451, ISBN 978-0-521-51597-9, Zbl1271.11073

• Agafonov, V. N. (1968), “Normal sequences and finite automata”, Soviet Mathematics Doklady 9: 324–325,Zbl 0242.94040.

• Bailey, D. H.; Crandall, R. E. (2001), “On the random character of fundamental constant expansions” (PDF),Experimental Mathematics 10: 175–190, doi:10.1080/10586458.2001.10504441.

• Bailey, D. H.; Crandall, R. E. (2002), “Random generators and normal numbers” (PDF), Experimental Math-ematics 11 (4): 527–546, doi:10.1080/10586458.2002.10504704.

• Bailey, D. H.; Misiurewicz, M. (2006), “A strong hot spot theorem”, Proceedings of the AmericanMathematicalSociety 134 (9): 2495–2501, doi:10.1090/S0002-9939-06-08551-0.

• Becher, V.; Figueira, S. (2002), “An example of a computable absolutely normal number”, Theoretical Com-puter Science 270: 947–958, doi:10.1016/S0304-3975(01)00170-0.

• Besicovitch, A. S. (1935), “The asymptotic distribution of the numerals in the decimal representation of thesquares of the natural numbers”, Mathematische Zeitschrift 39: 146–156, doi:10.1007/BF01201350.

• Borel, E. (1909), “Les probabilités dénombrables et leurs applications arithmétiques”, Rendiconti del CircoloMatematico di Palermo 27: 247–271, doi:10.1007/BF03019651.

• Bourke, C.; Hitchcock, J. M.; Vinodchandran, N. V. (2005), “Entropy rates and finite-state dimension”, The-oretical Computer Science 349 (3): 392–406, doi:10.1016/j.tcs.2005.09.040.

• Bugeaud, Yann (2012), Distribution modulo one and Diophantine approximation, Cambridge Tracts in Math-ematics 193, Cambridge: Cambridge University Press, ISBN 978-0-521-11169-0, Zbl pre06066616

• Calude, C. (1994), “Borel normality and algorithmic randomness”, in Rozenberg, G.; Salomaa, Arto,Developmentsin Language Theory: At the Crossroads of Mathematics, Computer Science and Biology, World Scientific, Sin-gapore, pp. 113–119.

• Calude, C.S.; Zamfirescu, T. (1999), “Most numbers obey no probability laws”, Publicationes MathematicaeDebrecen 54 (Supplement): 619–623.

42.8. FURTHER READING 89

• Cassels, J. W. S. (1959), “On a problem of Steinhaus about normal numbers”, Colloquium Mathematicum 7:95–101.

• Champernowne, D. G. (1933), “The construction of decimals normal in the scale of ten”, Journal of the LondonMathematical Society 8 (4): 254–260, doi:10.1112/jlms/s1-8.4.254.

• Copeland, A. H.; Erdős, P. (1946), “Note on normal numbers”, Bulletin of the American Mathematical Society52 (10): 857–860, doi:10.1090/S0002-9904-1946-08657-7.

• Dajani, Karma; Kraaikamp, Cor (2002), Ergodic theory of numbers, Carus Mathematical Monographs 29,Washington, DC: Mathematical Association of America, ISBN 0-88385-034-6, Zbl 1033.11040.

• Davenport, H.; Erdős, P. (1952), “Note on normal decimals”, Canadian Journal of Mathematics 4: 58–63,doi:10.4153/CJM-1952-005-3.

• Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003), Recurrence sequences, Mathe-matical Surveys andMonographs 104, Providence, RI: AmericanMathematical Society, ISBN 0-8218-3387-1,Zbl 1033.11006.

• Khoshnevisan, Davar (2006), “Normal numbers are normal” (PDF), Clay Mathematics Institute Annual Report2006: 15, continued pp. 27–31.

• Long, C. T. (1957), “Note on normal numbers”, Pacific Journal ofMathematics 7 (2): 1163–1165, doi:10.2140/pjm.1957.7.1163,Zbl 0080.03604.

• Martin, Greg (2001), “Absolutely abnormal numbers”,AmericanMathematicalMonthly 108: 746–754, doi:10.2307/2695618,Zbl 1036.11035

• Murty, Maruti Ram (2007), Problems in analytic number theory (2 ed.), Springer, ISBN 0-387-72349-8.• Nakai, Y.; Shiokawa, I. (1992), “Discrepancy estimates for a class of normal numbers”, Acta Arithmetica 62(3): 271–284.

• Schmidt,W. (1960), “On normal numbers”, Pacific Journal ofMathematics 10: 661–672, doi:10.2140/pjm.1960.10.661.• Schnorr, C. P.; Stimm, H. (1972), “Endliche Automaten und Zufallsfolgen”, Acta Informatica 1 (4): 345–359,doi:10.1007/BF00289514.

• Sierpiński, W. (1917), “Démonstration élémentaire d'un théorème de M. Borel sur les nombres absolutmentnormaux et détermination effective d'un tel nombre”, Bulletin de la Société Mathématique de France 45: 125–144.

• Wall, D. D. (1949), Normal Numbers, Ph.D. thesis, Berkeley, California: University of California.• Ziv, J.; Lempel, A. (1978), “Compression of individual sequences via variable-rate coding”, IEEE Transactions

on Information Theory 24 (5): 530–536, doi:10.1109/TIT.1978.1055934.

42.8 Further reading• Harman, Glyn (2002), “One hundred years of normal numbers”, in Bennett, M. A.; Berndt, B.C.; Boston, N.;Diamond, H.G.; Hildebrand, A.J.; Philipp,W., Surveys in number theory: Papers from the millennial conferenceon number theory, Natick, MA: A K Peters, pp. 57–74, ISBN 1-56881-162-4, Zbl 1062.11052

• Quéfflec, Martine (2006), “Old and new results on normality”, in Denteneer, Dee; den Hollander, F.; Ver-bitskiy, E., Dynamics & Stochastics: Festschrift in honor of M. S. Keane, IMS Lecture Notes – MonographSeries 48, Beachwood, Ohio: Institute of Mathematical Statistics, pp. 225–236, arXiv:math.DS/0608249,doi:10.1214/074921706000000248, ISBN 0-940600-64-1, Zbl 1130.11041

42.9 External links• We are in Digits of Pi and Live Forever by Clifford A. Pickover

• Weisstein, Eric W., “Normal number”, MathWorld.

Chapter 43

Perfect set property

In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has anonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as beinga perfect set.As nonempty perfect sets in a Polish space always have the cardinality of the continuum, a set with the perfect setproperty cannot be a counterexample to the continuum hypothesis, stated in the form that every uncountable set ofreals has the cardinality of the continuum.The Cantor–Bendixson theorem states that closed sets of a Polish space X have the perfect set property in a partic-ularly strong form; any closed set C may be written uniquely as the disjoint union of a perfect set P and a countableset S. Thus it follows that every closed subset of a Polish space has the perfect set property. In particular, everyuncountable Polish space has the perfect set property, and can be written as the disjoint union of a perfect set and acountable open set.It follows from the axiom of choice that there are sets of reals that do not have the perfect set property. Every analyticset has the perfect set property. It follows from sufficient large cardinals that every projective set has the perfect setproperty.

43.1 References• Kechris, A. S. (1995), Classical Descriptive Set Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94374-9

90

Chapter 44

Pointclass

In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a point isordinarily understood to be an element of some perfect Polish space. In practice, a pointclass is usually characterizedby some sort of definability property; for example, the collection of all open sets in some fixed collection of Polishspaces is a pointclass. (An open set may be seen as in some sense definable because it cannot be a purely arbitrarycollection of points; for any point in the set, all points sufficiently close to that point must also be in the set.)Pointclasses find application in formulating many important principles and theorems from set theory and real analysis.Strong set-theoretic principles may be stated in terms of the determinacy of various pointclasses, which in turn impliesthat sets in those pointclasses (or sometimes larger ones) have regularity properties such as Lebesgue measurability(and indeed universal measurability), the property of Baire, and the perfect set property.

44.1 Basic framework

In practice, descriptive set theorists often simplify matters by working in a fixed Polish space such as Baire spaceor sometimes Cantor space, each of which has the advantage of being zero dimensional, and indeed homeomorphicto its finite or countable powers, so that considerations of dimensionality never arise. Moschovakis provides greatergenerality by fixing once and for all a collection of underlying Polish spaces, including the set of all naturals, theset of all reals, Baire space, and Cantor space, and otherwise allowing the reader to throw in any desired perfectPolish space. Then he defines a product space to be any finite Cartesian product of these underlying spaces. Then, forexample, the pointclass Σ0

1 of all open sets means the collection of all open subsets of one of these product spaces.This approach prevents Σ0

1 from being a proper class, while avoiding excessive specificity as to the particular Polishspaces being considered (given that the focus is on the fact that Σ0

1 is the collection of open sets, not on the spacesthemselves).

44.2 Boldface pointclasses

The pointclasses in the Borel hierarchy, and in the more complex projective hierarchy, are represented by sub- andsuper-scripted Greek letters in boldface fonts; for example,Π0

1 is the pointclass of all closed sets,Σ02 is the pointclass

of all Fσ sets,∆02 is the collection of all sets that are simultaneously Fσ andGδ, andΣ1

1 is the pointclass of all analyticsets.Sets in such pointclasses need be “definable” only up to a point. For example, every singleton set in a Polish space isclosed, and thusΠ0

1 . Therefore, it cannot be that everyΠ01 set must be “more definable” than an arbitrary element

of a Polish space (say, an arbitrary real number, or an arbitrary countable sequence of natural numbers). Boldfacepointclasses, however, may (and in practice ordinarily do) require that sets in the class be definable relative to somereal number, taken as an oracle. In that sense, membership in a boldface pointclass is a definability property, eventhough it is not absolute definability, but only definability with respect to a possibly undefinable real number.Boldface pointclasses, or at least the ones ordinarily considered, are closed under Wadge reducibility; that is, given aset in the pointclass, its inverse image under a continuous function (from a product space to the space of which thegiven set is a subset) is also in the given pointclass. Thus a boldface pointclass is a downward-closed union of Wadge

91

92 CHAPTER 44. POINTCLASS

degrees.

44.3 Lightface pointclasses

The Borel and projective hierarchies have analogs in effective descriptive set theory in which the definability propertyis no longer relativized to an oracle, but is made absolute. For example, if one fixes some collection of basic openneighborhoods (say, in Baire space, the set of all sets of the form x∈ωω|x ⊇s for any fixed finite sequence s of naturalnumbers), then the open, or Σ0

1 , sets may be characterized as all (arbitrary) unions of basic open neighborhoods.The analogous Σ0

1 sets, with a lightface Σ , are no longer arbitrary unions of such neighborhoods, but computableunions of them (that is, a set is Σ0

1 if there is a computable set S of finite sequences of naturals such that the given setis the union of all x∈ωω|x ⊇s for s in S).A set is lightfaceΠ0

1 if it is the complement of a Σ01 set. Thus each Σ0

1 set has at least one index, which describes thecomputable function enumerating the basic open sets from which it is composed; in fact it will have infinitely manysuch indices. Similarly, an index for a Π0

1 set B describes the computable function enumerating the basic open setsin the complement of B.A set A is lightface Σ0

2 if it is a union of a computable sequence of Π01 sets (that is, there is a computable enu-

meration of indices of Π01 sets such that A is the union of these sets). This relationship between lightface sets and

their indices is used to extend the lightface Borel hierarchy into the transfinite, via recursive ordinals. This pro-duces that hyperarithmetic hierarchy, which is the lightface analog of the Borel hierarchy. (The finite levels of thehyperarithmetic hierarchy are known as the arithmetical hierarchy.)A similar treatment can be applied to the projective hierarchy. Its lightface analog is known as the analytical hierarchy.

44.4 References• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

Chapter 45

Polish group

In mathematics, a Polish group is a topological group that is also a Polish space, in other words homeomorphic to aseparable complete metric space.

45.1 Examples• All finite dimensional Lie groups with a countable number of components are Polish groups.

• The unitary group of a separable Hilbert space (with the strong topology) is a Polish group.

• The group of homeomorphisms of a compact metric space is a Polish group.

• The product of a countable number of Polish groups is a Polish group.

• The group of isometries of a separable complete metric space is a Polish group.

45.2 Properties

The group of homeomorphisms of the Hilbert cube [0,1]N is a universal Polish group, in the sense that every Polishgroup is isomorphic to a closed subgroup of it.

45.3 References• Kechris, A. (1995). Classical Descriptive Set Theory. Graduate Texts in Mathematics 156. Springer. ISBN0-387-94374-9.

93

Chapter 46

Polish space

In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topologicalspace; that is, a space homeomorphic to a completemetric space that has a countable dense subset. Polish spaces are sonamed because theywere first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarskiand others. However, Polish spaces are mostly studied today because they are the primary setting for descriptiveset theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for moreadvanced measure theory, in particular in probability theory.Common examples of Polish spaces are the real line, any separable Banach space, the Cantor space, and Baire space.Additionally, some spaces that are not complete metric spaces in the usual metric may be Polish; e.g., the open interval(0, 1) is Polish.Between any two uncountable Polish spaces, there is a Borel isomorphism; that is, a bijection that preserves the Borelstructure. In particular, every uncountable Polish space has the cardinality of the continuum.Lusin spaces, Suslin spaces, and Radon spaces are generalizations of Polish spaces.

46.1 Properties

1. (Alexandrov's theorem) If X is Polish then so is any Gδ subset of X.

2. (Cantor–Bendixson theorem) If X is Polish then any closed subset of X can be written as the disjoint union ofa perfect subset and a countable open subset.

3. A subspace Q of a Polish space P is Polish if and only if Q is the intersection of a sequence of open subsets ofP. (This is the converse to Alexandrov’s theorem.)

4. A topological space X is Polish if and only if X is homeomorphic to the intersection of a sequence of opensubsets of the cube IN , where I is the unit interval and N is the set of natural numbers.

The following spaces are Polish:

• closed subsets of a Polish space,

• open subsets of a Polish spaces

• products and disjoint unions of countable families of Polish spaces,

• locally compact spaces that are metrizable and countable at infinity,

• countable intersections of Polish subspaces of a Hausdorff topological space,

• the set of nonrational numbers with the topology induced by the real line.

94

46.2. CHARACTERIZATION 95

46.2 Characterization

There are numerous characterizations that tell when a second countable topological space is metrizable, such asUrysohn’s metrization theorem. The problem of determining whether a metrizable space is completely metrizableis more difficult. Topological spaces such as the open unit interval (0,1) can be given both complete metrics andincomplete metrics generating their topology.There is a characterization of complete separable metric spaces in terms of a game known as the strong Choquetgame. A separable metric space is completely metrizable if and only if the second player has a winning strategy inthis game.A second characterization follows from Alexandrov’s theorem. It states that a separable metric space is completelymetrizable if and only if it is a Gδ subset of its completion in the original metric.

46.3 Polish metric spaces

Although Polish spaces are metrizable, they are not in and of themselves metric spaces; each Polish space admitsmany complete metrics giving rise to the same topology, but no one of these is singled out or distinguished. A Polishspace with a distinguished complete metric is called a Polish metric space. An alternative approach, equivalent to theone given here, is first to define “Polish metric space” to mean “complete separable metric space”, and then to definea “Polish space” as the topological space obtained from a Polish metric space by forgetting the metric.

46.4 Generalizations of Polish spaces

46.4.1 Lusin spaces

A Lusin space is a topological space such that some weaker topology makes it into a Polish space.There are many ways to form Lusin spaces. In particular:

• Every Polish space is Lusin.

• A subspace of a Lusin space is Lusin if and only if it is a Borel set.

• Any countable union or intersection of Lusin subspaces of a Hausdorff space is Lusin.

• The product of a countable number of Lusin spaces is Lusin.

• The disjoint union of a countable number of Lusin spaces is Lusin.

46.4.2 Suslin spaces

A Suslin space is the image of a Polish space under a continuous mapping. So every Lusin space is Suslin. In aPolish space, a subset is a Suslin space if and only if it is a Suslin set (an image of the Suslin operation).The following are Suslin spaces:

• closed or open subsets of a Suslin space,

• countable products and disjoint unions of Suslin spaces,

• countable intersections or countable unions of Suslin subspaces of a Hausdorff topological space,

• continuous images of Suslin spaces,

• Borel subsets of a Suslin space.

They have the following properties:

• Every Suslin space is separable.

96 CHAPTER 46. POLISH SPACE

46.4.3 Radon spaces

Main article: Radon space

A Radon space is a topological space such that every finite Borel measure is inner regular (so a Radon measure).Every Suslin space is Radon.

46.4.4 Polish groups

A Polish group is a topological group G regarded as a topological space which is itself a Polish space. A remarkablefact about Polish groups is that Baire-measurable (i.e., the preimage of any open set has the property of Baire)homomorphisms between them are automatically continuous. (Pettis in B. J. Pettis, ‘On continuity and openness ofhomomorphisms in topological groups’, Ann. of Math. vol. 51 (1950) 293–308, MR 38358)

46.5 See also• Standard Borel space

46.6 References• Arveson, William (1981). An Invitation to C*-Algebras. Graduate Texts in Mathematics 39. New York:Springer-Verlag. ISBN 0-387-90176-0.

• Bourbaki, Nicolas (1966). Elements of Mathematics: General Topology. Addison–Wesley.

• Kechris, A. (1995). Classical Descriptive Set Theory. Graduate Texts in Mathematics 156. Springer. ISBN0-387-94374-9.

• Kuratowski, K. (1966). Topology Vol. I. Academic Press. ISBN 012429202X.

Chapter 47

Prewellordering

In set theory, a prewellordering is a binary relation ≤ that is transitive, total, and wellfounded (more precisely, therelation x ≤ y ∧ y ≰ x is wellfounded). In other words, if ≤ is a prewellordering on a setX , and if we define ∼ by

x ∼ y ⇐⇒ x ≤ y ∧ y ≤ x

then∼ is an equivalence relation onX , and≤ induces a wellordering on the quotientX/ ∼ . The order-type of thisinduced wellordering is an ordinal, referred to as the length of the prewellordering.A norm on a set X is a map from X into the ordinals. Every norm induces a prewellordering; if ϕ : X → Ord is anorm, the associated prewellordering is given by

x ≤ y ⇐⇒ ϕ(x) ≤ ϕ(y)

Conversely, every prewellordering is induced by a unique regular norm (a norm ϕ : X → Ord is regular if, for anyx ∈ X and any α < ϕ(x) , there is y ∈ X such that ϕ(y) = α ).

47.1 Prewellordering property

If Γ is a pointclass of subsets of some collection F of Polish spaces, F closed under Cartesian product, and if ≤ is aprewellordering of some subset P of some element X of F , then ≤ is said to be a Γ -prewellordering of P if therelations <∗ and ≤∗ are elements of Γ , where for x, y ∈ X ,

1. x <∗ y ⇐⇒ x ∈ P ∧ [y /∈ P ∨ x ≤ y ∧ y ≤ x]

2. x ≤∗ y ⇐⇒ x ∈ P ∧ [y /∈ P ∨ x ≤ y]

Γ is said to have the prewellordering property if every set in Γ admits a Γ -prewellordering.The prewellordering property is related to the stronger scale property; in practice, many pointclasses having theprewellordering property also have the scale property, which allows drawing stronger conclusions.

47.1.1 Examples

Π11 andΣ1

2 both have the prewellordering property; this is provable in ZFC alone. Assuming sufficient large cardinals,for every n ∈ ω ,Π1

2n+1 andΣ12n+2 have the prewellordering property.

47.1.2 Consequences

97

98 CHAPTER 47. PREWELLORDERING

Reduction

If Γ is an adequate pointclass with the prewellordering property, then it also has the reduction property: For anyspace X ∈ F and any sets A,B ⊆ X , A and B both in Γ , the union A ∪B may be partitioned into sets A∗, B∗ ,both in Γ , such that A∗ ⊆ A and B∗ ⊆ B .

Separation

If Γ is an adequate pointclass whose dual pointclass has the prewellordering property, then Γ has the separationproperty: For any space X ∈ F and any sets A,B ⊆ X , A and B disjoint sets both in Γ , there is a set C ⊆ Xsuch that both C and its complement X \ C are in Γ , with A ⊆ C and B ∩ C = ∅ .For example, Π1

1 has the prewellordering property, so Σ11 has the separation property. This means that if A and B

are disjoint analytic subsets of some Polish spaceX , then there is a Borel subset C ofX such that C includes A andis disjoint from B .

47.2 See also• Descriptive set theory

• Scale property

• Graded poset – a graded poset is analogous to a prewellordering with a norm, replacing a map to the ordinalswith a map to the integers

47.3 References• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

Chapter 48

Projective hierarchy

“Projective set” redirects here. For the card game, see Projective Set (game).

In the mathematical field of descriptive set theory, a subset A of a Polish space X is projective if it is Σ1n for some

positive integer n . Here A is

• Σ11 if A is analytic

• Π1n if the complement of A , X \A , is Σ1

n

• Σ1n+1 if there is a Polish space Y and a Π1

n subset C ⊆ X × Y such that A is the projection of C ; that is,A = x ∈ X | ∃y ∈ Y (x, y) ∈ C

The choice of the Polish space Y in the third clause above is not very important; it could be replaced in the definitionby a fixed uncountable Polish space, say Baire space or Cantor space or the real line.

48.1 Relationship to the analytical hierarchy

There is a close relationship between the relativized analytical hierarchy on subsets of Baire space (denoted by lightfaceletters Σ and Π ) and the projective hierarchy on subsets of Baire space (denoted by boldface lettersΣ andΠ ). Notevery Σ1

n subset of Baire space is Σ1n . It is true, however, that if a subset X of Baire space is Σ1

n then there isa set of natural numbers A such that X is Σ1,A

n . A similar statement holds for Π1n sets. Thus the sets classified

by the projective hierarchy are exactly the sets classified by the relativized version of the analytical hierarchy. Thisrelationship is important in effective descriptive set theory.A similar relationship between the projective hierarchy and the relativized analytical hierarchy holds for subsets ofCantor space and, more generally, subsets of any effective Polish space.

48.2 References• Kechris, A. S. (1995), Classical Descriptive Set Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94374-9

• Rogers, Hartley (1987) [1967], The Theory of Recursive Functions and Effective Computability, First MIT presspaperback edition, ISBN 978-0-262-68052-3

99

Chapter 49

Property of Baire

A subset A of a topological space X has the property of Baire (Baire property, named after René-Louis Baire),or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set U ⊆ Xsuch that A∆U is meager (where Δ denotes the symmetric difference).[1]

The family of sets with the property of Baire forms a σ-algebra. That is, the complement of an almost open set isalmost open, and any countable union or intersection of almost open sets is again almost open.[1] Since every openset is almost open (the empty set is meager), it follows that every Borel set is almost open.If a subset of a Polish space has the property of Baire, then its corresponding Banach-Mazur game is determined.The converse does not hold; however, if every game in a given adequate pointclass Γ is determined, then every set inΓ has the property of Baire. Therefore it follows from projective determinacy, which in turn follows from sufficientlarge cardinals, that every projective set (in a Polish space) has the property of Baire.[2]

It follows from the axiom of choice that there are sets of reals without the property of Baire. In particular, the Vitaliset does not have the property of Baire.[3] Already weaker versions of choice are sufficient: the Boolean prime idealtheorem implies that there is a nonprincipal ultrafilter on the set of natural numbers; each such ultrafilter induces, viabinary representations of reals, a set of reals without the Baire property.[4]

49.1 See also• Baire category theorem

49.2 References[1] Oxtoby, John C. (1980), “4. The Property of Baire”, Measure and Category, Graduate Texts in Mathematics 2 (2nd ed.),

Springer-Verlag, pp. 19–21, ISBN 978-0-387-90508-2.

[2] Becker, Howard; Kechris, Alexander S. (1996), The descriptive set theory of Polish group actions, London MathematicalSociety Lecture Note Series 232, Cambridge University Press, Cambridge, p. 69, doi:10.1017/CBO9780511735264,ISBN 0-521-57605-9, MR 1425877.

[3] Oxtoby (1980), p. 22.

[4] Blass, Andreas (2010), “Ultrafilters and set theory”,Ultrafilters across mathematics, ContemporaryMathematics 530, Prov-idence, RI: American Mathematical Society, pp. 49–71, doi:10.1090/conm/530/10440, MR 2757533. See in particularp. 64.

49.3 External links• Springer Encyclopaedia of Mathematics article on Baire property

100

Chapter 50

Scale (descriptive set theory)

In the mathematical discipline of descriptive set theory, a scale is a certain kind of object defined on a set of pointsin some Polish space (for example, a scale might be defined on a set of real numbers). Scales were originally isolatedas a concept in the theory of uniformization,[1] but have found wide applicability in descriptive set theory, withapplications such as establishing bounds on the possible lengths of wellorderings of a given complexity, and showing(under certain assumptions) that there are largest countable sets of certain complexities.

50.1 Formal definition

Given a pointset A contained in some product space

A ⊆ X = X0 ×X1 × . . . Xm−1

where each Xk is either the Baire space or a countably infinite discrete set, we say that a norm on A is a map fromA into the ordinal numbers. Each norm has an associated prewellordering, where one element of A precedes anotherelement if the norm of the first is less than the norm of the second.A scale on A is a countably infinite collection of norms

(ϕn)n<ω

with the following properties:

If the sequence xi is such that

xi is an element of A for each natural number i, andxi converges to an element xin the product space X, andfor each natural number n there is an ordinal λn such that φ (xi)=λn for all sufficiently largei, then

x is an element of A, andfor each n, φ (x)≤λn.[2]

By itself, at least granted the axiom of choice, the existence of a scale on a pointset is trivial, as A can be wellorderedand each φn can simply enumerate A. To make the concept useful, a definability criterion must be imposed on thenorms (individually and together). Here “definability” is understood in the usual sense of descriptive set theory; itneed not be definability in an absolute sense, but rather indicates membership in some pointclass of sets of reals. Thenorms φn themselves are not sets of reals, but the corresponding prewellorderings are (at least in essence).The idea is that, for a given pointclass Γ, we want the prewellorderings below a given point in A to be uniformlyrepresented both as a set in Γ and as one in the dual pointclass of Γ, relative to the “larger” point being an element of

101

102 CHAPTER 50. SCALE (DESCRIPTIVE SET THEORY)

A. Formally, we say that the φn form a Γ-scale on A if they form a scale on A and there are ternary relations S andT such that, if y is an element of A, then

∀n∀x(φn(x) ≤ φn(y) ⇐⇒ S(n, x, y) ⇐⇒ T (n, x, y))

where S is in Γ and T is in the dual pointclass of Γ (that is, the complement of T is in Γ).[3] Note here that we thinkof φn(x) as being ∞ whenever x∉A; thus the condition φn(x)≤φn(y), for y∈A, also implies x∈A.Note also that the definition does not imply that the collection of norms is in the intersection of Γ with the dualpointclass of Γ. This is because the three-way equivalence is conditional on y being an element of A. For y not in A,it might be the case that one or both of S(n,x,y) or T(n,x,y) fail to hold, even if x is in A (and therefore automaticallyφn(x)≤φn(y)=∞).

50.2 ApplicationsThis section is yet to be written

50.3 Scale property

The scale property is a strengthening of the prewellordering property. For pointclasses of a certain form, it impliesthat relations in the given pointclass have a uniformization that is also in the pointclass.

50.4 PeriodicityThis section is yet to be written

50.5 Notes[1] Kechris and Moschovakis 2008:28

[2] Kechris and Moschovakis 2008:37

[3] Kechris and Moschovakis 2008:37, with harmless reworking

50.6 References• Moschovakis, Yiannis N. (1980), Descriptive Set Theory, North Holland, ISBN 0-444-70199-0

• Kechris, Alexander S.; Moschovakis, Yiannis N. (2008), “Notes on the theory of scales”, in Kechris, Alexan-der S.; Benedikt Löwe; Steel, John R., Games, Scales and Suslin Cardinals: The Cabal Seminar, Volume I,Cambridge University Press, pp. 28–74, ISBN 978-0-521-89951-2

Chapter 51

Schröder–Bernstein theorem formeasurable spaces

The Cantor–Bernstein–Schroeder theorem of set theory has a counterpart for measurable spaces, sometimes calledtheBorel Schroeder–Bernstein theorem, sincemeasurable spaces are also called Borel spaces. This theorem, whoseproof is quite easy, is instrumental when proving that two measurable spaces are isomorphic. The general theory ofstandard Borel spaces contains very strong results about isomorphic measurable spaces, see Kuratowski’s theorem.However, (a) the latter theorem is very difficult to prove, (b) the former theorem is satisfactory in many importantcases (see Examples), and (c) the former theorem is used in the proof of the latter theorem.

51.1 The theorem

Let X and Y be measurable spaces. If there exist injective, bimeasurable maps f : X → Y, g : Y → X, then Xand Y are isomorphic (the Schröder–Bernstein property).

51.1.1 Comments

The phrase " f is bimeasurable” means that, first, f is measurable (that is, the preimage f−1(B) is measurable forevery measurableB ⊂ Y ), and second, the image f(A) is measurable for every measurableA ⊂ X . (Thus, f(X)must be a measurable subset of Y, not necessarily the whole Y. )An isomorphism (between two measurable spaces) is, by definition, a bimeasurable bijection. If it exists, thesemeasurable spaces are called isomorphic.

51.1.2 Proof

First, one constructs a bijection h : X → Y out of f and g exactly as in the proof of the Cantor–Bernstein–Schroeder theorem. Second, h is measurable, since it coincides with f on a measurable set and with g−1 on itscomplement. Similarly, h−1 is measurable.

51.2 Examples

51.2.1 Example 1

The open interval (0, 1) and the closed interval [0, 1] are evidently non-isomorphic as topological spaces (that is,not homeomorphic). However, they are isomorphic as measurable spaces. Indeed, the closed interval is evidentlyisomorphic to a shorter closed subinterval of the open interval. Also the open interval is evidently isomorphic to apart of the closed interval (just itself, for instance).

103

104 CHAPTER 51. SCHRÖDER–BERNSTEIN THEOREM FOR MEASURABLE SPACES

Example maps f:(0,1)→[0,1] and g:[0,1]→(0,1).

51.2.2 Example 2

The real line R and the plane R2 are isomorphic as measurable spaces. It is immediate to embed R into R2. Theconverse, embedding of R2. into R (as measurable spaces, of course, not as topological spaces) can be made by awell-known trick with interspersed digits; for example,

g(π,100e) = g(3.14159 265…, 271.82818 28…) = 20731.184218 51982 2685….

The map g : R2 → R is clearly injective. It is easy to check that it is bimeasurable. (However, it is not bijective; forexample, the number 1/11 = 0.090909 . . . is not of the form g(x, y) ).

51.3 References• S.M. Srivastava, A Course on Borel Sets, Springer, 1998.

See Proposition 3.3.6 (on page 96), and the first paragraph of Section 3.3 (on page 94).

Chapter 52

Smith–Volterra–Cantor set

After black intervals have been removed, the white points which remain are a nowhere dense set of measure 1/2.

In mathematics, the Smith–Volterra–Cantor set (SVC), fat Cantor set, or ε-Cantor set[1] is an example of a setof points on the real lineR that is nowhere dense (in particular it contains no intervals), yet has positive measure. TheSmith–Volterra–Cantor set is named after the mathematicians Henry Smith, Vito Volterra and Georg Cantor. TheSmith-Volterra-Cantor set is topologically equivalent to the middle-thirds Cantor set.

52.1 Construction

Similar to the construction of the Cantor set, the Smith–Volterra–Cantor set is constructed by removing certainintervals from the unit interval [0, 1].The process begins by removing the middle 1/4 from the interval [0, 1] (the same as removing 1/8 on either side ofthe middle point at 1/2) so the remaining set is

[0,

3

8

]∪[5

8, 1

].

The following steps consist of removing subintervals of width 1/22n from the middle of each of the 2n−1 remainingintervals. So for the second step the intervals (5/32, 7/32) and (25/32, 27/32) are removed, leaving

[0,

5

32

]∪[7

32,3

8

]∪[5

8,25

32

]∪[27

32, 1

].

Continuing indefinitely with this removal, the Smith–Volterra–Cantor set is then the set of points that are neverremoved. The image below shows the initial set and five iterations of this process.Each subsequent iterate in the Smith–Volterra–Cantor set’s construction removes proportionally less from the re-maining intervals. This stands in contrast to the Cantor set, where the proportion removed from each interval remainsconstant. Thus, the former has positive measure, while the latter zero measure.

105

106 CHAPTER 52. SMITH–VOLTERRA–CANTOR SET

52.2 Properties

By construction, the Smith–Volterra–Cantor set contains no intervals and therefore has empty interior. It is also theintersection of a sequence of closed sets, which means that it is closed. During the process, intervals of total length

∞∑n=0

2n

22n+2=

1

4+

1

8+

1

16+ · · · = 1

2

are removed from [0, 1], showing that the set of the remaining points has a positive measure of 1/2. This makes theSmith–Volterra–Cantor set an example of a closed set whose boundary has positive Lebesgue measure.

52.3 Other fat Cantor sets

In general, one can remove rn from each remaining subinterval at the n-th step of the algorithm, and end up with aCantor-like set. The resulting set will have positive measure if and only if the sum of the sequence is less than themeasure of the initial interval.Cartesian products of Smith–Volterra–Cantor sets can be used to find totally disconnected sets in higher dimensionswith nonzero measure. By applying the Denjoy–Riesz theorem to a two-dimensional set of this type, it is possible tofind a Jordan curve such that the points on the curve have positive area.[2]

52.4 See also• The SVC is used in the construction of Volterra’s function (see external link).

• The SVC is an example of a compact set that is not Jordan measurable, see Jordan measure#Extension to morecomplicated sets.

• The indicator function of the SVC is an example of a bounded function that is not Riemann integrable on(0,1) and moreover, is not equal almost everywhere to a Riemann integrable function, see Riemann inte-gral#Examples.

52.5 References[1] Aliprantis and Burkinshaw (1981), Principles of Real Analysis

[2] Balcerzak, M.; Kharazishvili, A. (1999), “On uncountable unions and intersections of measurable sets”, Georgian Mathe-matical Journal 6 (3): 201–212, doi:10.1023/A:1022102312024, MR 1679442.

52.6 External links• Wrestling with the Fundamental Theorem of Calculus: Volterra’s function, talk by David Marius Bressoud

Chapter 53

Stoneham number

In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G.Stoneham (1920–1996). For coprime numbers b, c > 1, the Stoneham number αb,c is defined as

αb,c =∑

n=ck>1

1

bnn=

∞∑k=1

1

bckck

It was shown by Stoneham in 1973 that αb,c is b-normal whenever c is an odd prime and b is a primitive root of c2.

53.1 References• Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Math-ematics 193. Cambridge: Cambridge University Press. ISBN 978-0-521-11169-0. Zbl pre06066616.

• Stoneham, R.G. (1973). “On absolute $(j,ε)$-normality in the rational fractions with applications to normalnumbers”. Acta Arithmetica 22: 277–286. Zbl 0276.10028.

• Stoneham, R.G. (1973). “On the uniform ε-distribution of residues within the periods of rational fractionswith applications to normal numbers”. Acta Arithmetica 22: 371–389. Zbl 0276.10029.

107

Chapter 54

Suslin operation

In mathematics, the Suslin operation A is an operation that constructs a set from a collection of sets indexed byfinite sequences of positive integers. The Suslin operation was introduced by Alexandrov (1916) and Suslin (1917).In Russia it is sometimes called the A-operation after Alexandrov. It is sometimes denoted by the symbol A (acalligraphic capital letter A).

54.1 Definitions

Suppose we have Suslin scheme, in other words a function M from finite sequences of positive integers n1,...,nk tosets Mn1,...,nk. The result of the Suslin operation is the set

A(M) = ∪ (Mn1 ∩ Mn1,n2 ∩ Mn1,n2, n3 ∩ ...)

where the union is taken over all infinite sequences n1,...,nk,...IfM is a family of subsets of a setX, thenA(M) is the family of subsets ofX obtained by applying the Suslin operationA to all collections as above where all the sets Mn1,...,nk are inM. The Suslin operation on collections of subsets ofX has the property that A(A(M)) = A(M). The family A(M) is closed under taking countable unions or intersections,but is not in general closed under taking complements.If M is the family of closed subsets of a topological space, then the elements of A(M) are called Suslin sets, oranalytic sets if the space is a Polish space.

54.2 References• Aleksandrov, P.S. (1916), C.R. Acad. Sci. Paris 162: 323–325 Missing or empty |title= (help)

• Hazewinkel, Michiel, ed. (2001), “A-operation”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Suslin, M.Ya. (1917), C.R. Acad. Sci. Paris 164: 88–91 Missing or empty |title= (help)

108

Chapter 55

Tree (descriptive set theory)

This article is about mathematical trees described by prefixes of finite sequences. For trees described by partiallyordered sets, see Tree (set theory).

In descriptive set theory, a tree on a setX is a collection of finite sequences of elements ofX such that every prefixof a sequence in the collection also belongs to the collection.

55.1 Definitions

55.1.1 Trees

The collection of all finite sequences of elements of a setX is denotedX<ω . With this notation, a tree is a nonemptysubset T of X<ω , such that if ⟨x0, x1, . . . , xn−1⟩ is a sequence of length n in T , and if 0 ≤ m < n , then theshortened sequence ⟨x0, x1, . . . , xm−1⟩ also belongs to T . In particular, choosing m = 0 shows that the emptysequence belongs to every tree.

55.1.2 Branches and bodies

A branch through a tree T is an infinite sequence of elements ofX , each of whose finite prefixes belongs to T . Theset of all branches through T is denoted [T ] and called the body of the tree T .A tree that has no branches is called wellfounded; a tree with at least one branch is illfounded. By König’s lemma, atree on a finite set with an infinite number of sequences must necessarily be illfounded.

55.1.3 Terminal nodes

A finite sequence that belongs to a tree T is called a terminal node if it is not a prefix of a longer sequence in T .Equivalently, ⟨x0, x1, . . . , xn−1⟩ ∈ T is terminal if there is no element x ofX such that that ⟨x0, x1, . . . , xn−1, x⟩ ∈T . A tree that does not have any terminal nodes is called pruned.

55.2 Relation to other types of trees

In graph theory, a rooted tree is a directed graph in which every vertex except for a special root vertex has exactlyone outgoing edge, and in which the path formed by following these edges from any vertex eventually leads to theroot vertex. If T is a tree in the descriptive set theory sense, then it corresponds to a graph with one vertex for eachsequence in T , and an outgoing edge from each nonempty sequence that connects it to the shorter sequence formedby removing its last element. This graph is a tree in the graph-theoretic sense. The root of the tree is the emptysequence.

109

110 CHAPTER 55. TREE (DESCRIPTIVE SET THEORY)

In order theory, a different notion of a tree is used: an order-theoretic tree is a partially ordered set with one minimalelement in which each element has a well-ordered set of predecessors. Every tree in descriptive set theory is also anorder-theoretic tree, using a partial ordering in which two sequences T and U are ordered by T < U if and only ifT is a proper prefix of U . The empty sequence is the unique minimal element, and each element has a finite andwell-ordered set of predecessors (the set of all of its prefixes). An order-theoretic tree may be represented by anisomorphic tree of sequences if and only if each of its elements has finite height (that is, a finite set of predecessors).

55.3 Topology

The set of infinite sequences over X (denoted as Xω ) may be given the product topology, treating X as a discretespace. In this topology, every closed subset C of Xω is of the form [T ] for some pruned tree T . Namely, let Tconsist of the set of finite prefixes of the infinite sequences in C . Conversely, the body [T ] of every tree T forms aclosed set in this topology.Frequently trees on Cartesian productsX ×Y are considered. In this case, by convention, the set of finite sequencesof members of the product space, (X × Y )<ω , is identified in the natural way with a subset of the product of twospaces of sequences,X<ω × Y <ω (the subset of members of the second product for which both sequences have thesame length). In this way a tree [T ] over the product space may be considered as a subset ofX<ω × Y <ω . We maythen form the projection of [T ] ,

p[T ] = x ∈ Xω|(∃y ∈ Y ω)⟨x, y⟩ ∈ [T ]

55.4 See also• Laver tree, a type of tree used in set theory as part of a notion of forcing

55.5 References• Kechris, Alexander S. (1995). Classical Descriptive Set Theory. Graduate Texts in Mathematics 156. Springer.ISBN 0-387-94374-9 ISBN 3-540-94374-9.

Chapter 56

Uniformization (set theory)

In set theory, the axiom of uniformization, a weak form of the axiom of choice, states that ifR is a subset ofX×Y, where X and Y are Polish spaces, then there is a subset f of R that is a partial function from X to Y , and whosedomain (in the sense of the set of all x such that f(x) exists) equals

x ∈ X|∃y ∈ Y (x, y) ∈ R

Such a function is called a uniformizing function for R , or a uniformization of R .

Uniformization of relation R (light blue) by function f (red).

To see the relationship with the axiom of choice, observe that R can be thought of as associating, to each elementof X , a subset of Y . A uniformization of R then picks exactly one element from each such subset, whenever thesubset is nonempty. Thus, allowing arbitrary sets X and Y (rather than just Polish spaces) would make the axiom ofuniformization equivalent to AC.

111

112 CHAPTER 56. UNIFORMIZATION (SET THEORY)

A pointclass Γ is said to have the uniformization property if every relation R in Γ can be uniformized by a partialfunction in Γ . The uniformization property is implied by the scale property, at least for adequate pointclasses of acertain form.It follows from ZFC alone thatΠ1

1 andΣ12 have the uniformization property. It follows from the existence of sufficient

large cardinals that

• Π12n+1 andΣ1

2n+2 have the uniformization property for every natural number n .

• Therefore, the collection of projective sets has the uniformization property.

• Every relation in L(R) can be uniformized, but not necessarily by a function in L(R). In fact, L(R) does nothave the uniformization property (equivalently, L(R) does not satisfy the axiom of uniformization).

• (Note: it’s trivial that every relation in L(R) can be uniformized in V, assuming V satisfies AC. The pointis that every such relation can be uniformized in some transitive inner model of V in which AD holds.)

56.1 References• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

Chapter 57

Unit interval

For the data transmission signaling interval, see Unit interval (data transmission).In mathematics, the unit interval is the closed interval [0,1], that is, the set of all real numbers that are greater than

0 1The unit interval as a subset of the real line

or equal to 0 and less than or equal to 1. It is often denoted I (capital letter I). In addition to its role in real analysis,the unit interval is used to study homotopy theory in the field of topology.In the literature, the term “unit interval” is sometimes applied to the other shapes that an interval from 0 to 1 couldtake: (0,1], [0,1), and (0,1). However, the notation I is most commonly reserved for the closed interval [0,1].

57.1 Properties

The unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological spaceit is compact, contractible, path connected and locally path connected. The Hilbert cube is obtained by taking atopological product of countably many copies of the unit interval.In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of thetwo points 0 and 1. Its standard orientation goes from 0 to 1.The unit interval is a totally ordered set and a complete lattice (every subset of the unit interval has a supremum andan infimum).

57.1.1 Cardinality

Main article: Cardinality of the continuum

The size or cardinality of a set is the number of elements it contains.The unit interval is a subset of the real numbers R . However, it has the same size as the whole set: the cardinality ofthe continuum. Since the real numbers can be used to represent points along an infinitely long line, this implies thata line segment of length 1, which is a part of that line, has the same number of points as the whole line. Moreover, ithas the same number of points as a square of area 1, as a cube of volume 1, and even as an unbounded n-dimensionalEuclidean space Rn (see Space filling curve).

113

114 CHAPTER 57. UNIT INTERVAL

The number of elements (either real numbers or points) in all the above-mentioned sets is uncountable, as it is strictlygreater than the number of natural numbers.

57.2 Generalizations

The interval [−1,1], with length two, demarcated by the positive and negative units, occurs frequently, such as in therange of the trigonometric functions sine and cosine and the hyperbolic function tanh. This interval may be used forthe domain of inverse functions. For instance, when θ is restricted to [−π/2, π/2] then sin(θ) is in this interval andarcsine is defined there.Sometimes, the term “unit interval” is used to refer to objects that play a role in various branches of mathematicsanalogous to the role that [0,1] plays in homotopy theory. For example, in the theory of quivers, the (analogue of the)unit interval is the graph whose vertex set is 0,1 and which contains a single edge e whose source is 0 and whosetarget is 1. One can then define a notion of homotopy between quiver homomorphisms analogous to the notion ofhomotopy between continuous maps.

57.3 Fuzzy logic

In logic, the unit interval [0,1] can be interpreted as a generalization of the Boolean domain 0,1, in which caserather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically,negation (NOT) is replaced with 1 − x ; conjunction (AND) is replaced with multiplication ( xy ); and disjunction(OR) is defined, per De Morgan’s laws, as 1− (1− x)(1− y) .Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logicand probabilistic logic. In these interpretations, a value is interpreted as the “degree” of truth – to what extent aproposition is true, or the probability that the proposition is true.

57.4 See also• Interval notation

• Unit square, cube, circle, hyperbola and sphere

• Unit vector

57.5 References• Robert G. Bartle, 1964, The Elements of Real Analysis, John Wiley & Sons.

Chapter 58

Universally Baire set

In the mathematical field of descriptive set theory, a set of real numbers (or more generally a subset of the Baire spaceor Cantor space) is called universally Baire if it has a certain strong regularity property. Universally Baire sets playan important role in Ω-logic, a very strong logical system invented by W. Hugh Woodin and the centerpiece of hisargument against the continuum hypothesis of Georg Cantor.

58.1 Definition

A subset A of the Baire space is universally Baire if it has one of the following equivalent properties:

1. For every notion of forcing, there are trees T and U such that A is the projection of the set of all branchesthrough T, and it is forced that the projections of the branches through T and the branches through U arecomplements of each other.

2. For every compact Hausdorff space Ω, and every continuous function f fromΩ to the Baire space, the preimageof A under f has the property of Baire in Ω.

3. For every cardinal λ and every continuous function f from λω to the Baire space, the preimage of A under fhas the property of Baire.

58.2 References• Bagaria, Joan; Todorcevic, Stevo (eds.). Set Theory: Centre de Recerca Matemàtica Barcelona, 2003-2004.Trends in Mathematics. ISBN 978-3-7643-7691-8.

• Feng, Qi; Magidor, Menachem; Woodin, Hugh. Judah, H.; Just, W.; Woodin, Hugh, eds. Set Theory of theContinuum. Mathematical Sciences Research Institute Publications.

115

Chapter 59

Universally measurable set

In mathematics, a subset A of a Polish space X is universally measurable if it is measurable with respect to everycomplete probability measure onX that measures all Borel subsets ofX . In particular, a universally measurable setof reals is necessarily Lebesgue measurable (see #Finiteness condition) below.Every analytic set is universally measurable. It follows from projective determinacy, which in turn follows fromsufficient large cardinals, that every projective set is universally measurable.

59.1 Finiteness condition

The condition that the measure be a probability measure; that is, that the measure of X itself be 1, is less restrictivethan it may appear. For example, Lebesgue measure on the reals is not a probability measure, yet every universallymeasurable set is Lebesgue measurable. To see this, divide the real line into countably many intervals of length 1; say,N0=[0,1), N1=[1,2), N2=[−1,0), N3=[2,3), N4=[−2,−1), and so on. Now letting μ be Lebesgue measure, define anew measure ν by

ν(A) =∞∑i=0

1

2n+1µ(A ∩Ni)

Then easily ν is a probability measure on the reals, and a set is ν-measurable if and only if it is Lebesgue measurable.More generally a universally measurable set must be measurable with respect to every sigma-finite measure thatmeasures all Borel sets.

59.2 Example contrasting with Lebesgue measurability

Suppose A is a subset of Cantor space 2ω ; that is, A is a set of infinite sequences of zeroes and ones. By putting abinary point before such a sequence, the sequence can be viewed as a real number between 0 and 1 (inclusive), withsome unimportant ambiguity. Thus we can think of A as a subset of the interval [0,1], and evaluate its Lebesguemeasure. That value is sometimes called the coin-flipping measure of A , because it is the probability of producinga sequence of heads and tails that is an element of A , upon flipping a fair coin infinitely many times.Now it follows from the axiom of choice that there are some such A without a well-defined Lebesgue measure (orcoin-flipping measure). That is, for such an A , the probability that the sequence of flips of a fair coin will wind upin A is not well-defined. This is a pathological property of A that says that A is “very complicated” or “ill-behaved”.From such a set A , form a new set A′ by performing the following operation on each sequence in A : Interspersea 0 at every even position in the sequence, moving the other bits to make room. Now A′ is intuitively no “simpler”or “better-behaved” than A . However, the probability that the sequence of flips of a fair coin will wind up in A′ iswell-defined, for the rather silly reason that the probability is zero (to get into A′ , the coin must come up tails onevery even-numbered flip).

116

59.3. REFERENCES 117

For such a set of sequences to be universallymeasurable, on the other hand, an arbitrarily biased coin may be used—even one that can “remember” the sequence of flips that has gone before—and the probability that the sequence ofits flips ends up in the set, must be well-defined. Thus the A′ described above is not universally measurable, becausewe can test it against a coin that always comes up tails on even-numbered flips, and is fair on odd-numbered flips.

59.3 References• Alexander Kechris (1995), Classical Descriptive Set Theory, Graduate Texts in Mathematics 156, Springer,ISBN 0-387-94374-9

• Nishiura Togo (2008), Absolute Measurable Spaces, Cambridge University Press, ISBN 0-521-87556-0

Chapter 60

Vitali set

In mathematics, aVitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, foundby Giuseppe Vitali.[1] The Vitali theorem is the existence theorem that there are such sets. There are uncountablymany Vitali sets, and their existence is proven on the assumption of the axiom of choice.

60.1 Measurable sets

Certain sets have a definite 'length' or 'mass’. For instance, the interval [0, 1] is deemed to have length 1; moregenerally, an interval [a, b], a ≤ b, is deemed to have length b−a. If we think of such intervals as metal rods withuniform density, they likewise have well-defined masses. The set [0, 1] ∪ [2, 3] is composed of two intervals of lengthone, so we take its total length to be 2. In terms of mass, we have two rods of mass 1, so the total mass is 2.There is a natural question here: if E is an arbitrary subset of the real line, does it have a 'mass’ or 'total length'? Asan example, we might ask what is the mass of the set of rational numbers, given that the mass of the interval [0, 1]is 1. The rationals are dense in the reals, so any non negative value may appear reasonable.However the closest generalization to mass is sigma additivity, which gives rise to the Lebesgue measure. It assignsa measure of b − a to the interval [a, b], but will assign a measure of 0 to the set of rational numbers because itis countable. Any set which has a well-defined Lebesgue measure is said to be “measurable”, but the constructionof the Lebesgue measure (for instance using Carathéodory’s extension theorem) does not make it obvious whethernon-measurable sets exist. The answer to that question involves the axiom of choice.

60.2 Construction and proof

A Vitali set is a subset V of the interval [0, 1] of real numbers such that, for each real number r, there is exactly onenumber v ∈ V such that v−r is a rational number. Vitali sets exist because the rational numbers Q form a normalsubgroup of the real numbersR under addition, and this allows the construction of the additive quotient groupR/Q ofthese two groups which is the group formed by the cosets of the rational numbers as a subgroup of the real numbersunder addition. This group R/Q consists of disjoint “shifted copies” of the rational numbers in the sense that eachelement of this quotient group is a set of the form Q + r for some r in R. The uncountably many elements of R/QpartitionR, and each element is dense inR. Each element ofR/Q intersects [0, 1], and the axiom of choice guaranteesthe existence of a subset of [0, 1] containing exactly one representative out of each element of R/Q. A set formedthis way is called a Vitali set.Every Vitali set V is uncountable, and v−u is irrational for any u, v ∈ V, u = v .A Vitali set is non-measurable. To show this, we assume that V is measurable and we derive a contradiction. Let q1,q2, ... be an enumeration of the rational numbers in [−1, 1] (recall that the rational numbers are countable). Fromthe construction of V, note that the translated sets Vk = V + qk = v + qk : v ∈ V , k = 1, 2, ... are pairwisedisjoint, and further note that

118

60.3. SEE ALSO 119

[0, 1] ⊆∪k

Vk ⊆ [−1, 2]

To see the first inclusion, consider any real number r in [0, 1] and let v be the representative in V for the equivalenceclass [r]; then r-v=qᵢ for some rational number qᵢ in [−1, 1] which implies that r is in V ᵢ.Apply the Lebesgue measure to these inclusions using sigma additivity:

1 ≤∞∑k=1

λ(Vk) ≤ 3.

Because the Lebesgue measure is translation invariant, λ(Vk) = λ(V ) and therefore

1 ≤∞∑k=1

λ(V ) ≤ 3.

But this is impossible. Summing infinitely many copies of the constant λ(V) yields either zero or infinity, accordingto whether the constant is zero or positive. In neither case is the sum in [1, 3]. So V cannot have been measurableafter all, i.e., the Lebesgue measure λ must not define any value for λ(V).

60.3 See also• Non-measurable set

• Banach–Tarski paradox

60.4 References[1] Vitali, Giuseppe (1905). “Sul problema della misura dei gruppi di punti di una retta”. Bologna, Tip. Gamberini e Parmeg-

giani.

60.5 Bibliography• Herrlich, Horst (2006). Axiom of Choice. Springer. p. 120.

• Vitali, Giuseppe (1905). “Sul problema della misura dei gruppi di punti di una retta”. Bologna, Tip. Gamberinie Parmeggiani.

Chapter 61

Wadge hierarchy

In descriptive set theory,Wadge degrees are levels of complexity for sets of reals. Sets are compared by continuousreductions. TheWadge hierarchy is the structure of Wadge degrees.

61.1 Wadge degrees

Suppose A and B are subsets of Baire space ωω. A isWadge reducible to B or A ≤W B if there is a continuousfunction f on ωω with A = f−1[B] . TheWadge order is the preorder or quasiorder on the subsets of Baire space.Equivalence classes of sets under this preorder are called Wadge degrees, the degree of a set A is denoted by [ A]W. The set of Wadge degrees ordered by the Wadge order is called theWadge hierarchy.Properties of Wadge degrees include their consistency with measures of complexity stated in terms of definability.For example, if A ≤W B and B is a countable intersection of open sets, then so is A . The same works for all levelsof the Borel hierarchy and the difference hierarchy. The Wadge hierarchy plays an important role in models of theaxiom of determinacy. Further interest in Wadge degrees comes from computer science, where some papers havesuggested Wadge degrees are relevant to algorithmic complexity.

61.2 Wadge and Lipschitz games

TheWadge game is a simple infinite game discovered by William Wadge (pronounced “wage”). It is used to inves-tigate the notion of continuous reduction for subsets of Baire space. Wadge had analyzed the structure of the Wadgehierarchy for Baire space with games by 1972, but published these results only much later in his PhD thesis. In theWadge game G(A,B) , player I and player II each in turn play integers which may depend on those played before.The outcome of the game is determined by checking whether the sequences x and y generated by players I and II arecontained in the sets A and B, respectively. Player II wins if the outcome is the same for both players, i.e. x is in Aif and only if y is in B . Player I wins if the outcome is different. Sometimes this is also called the Lipschitz game,and the variant where player II has the option to pass (but has to play infinitely often) is called the Wadge game.Suppose for a moment that the game is determined. If player I has a winning strategy, then this defines a continuous(even Lipschitz) map reducing B to the complement of A , and if on the other hand player II has a winning strategythen you have a reduction ofA toB . For example, suppose that player II has a winning strategy. Map every sequencex to the sequence y that player II plays inG(A,B) if player I plays the sequence x, where player II follows his or herwinning strategy. This defines a is a continuous map f with the property that x is in A if and only if f(x) is in B .Wadge’s lemma states that under the axiom of determinacy (AD), for any two subsets A,B of Baire space, A ≤WB or B ≤W ωω– A . The assertion that the Wadge lemma holds for sets in Γ is the semilinear ordering principle forΓ or SLO(Γ). Any semilinear order defines a linear order on the equivalence classes modulo complements. Wadge’slemma can be applied locally to any pointclass Γ, for example the Borel sets,Δ1 sets, Σ1 sets, orΠ1 sets. It followsfrom determinacy of differences of sets in Γ. Since Borel determinacy is proved in ZFC, ZFC implies Wadge’s lemmafor Borel sets.

120

61.3. STRUCTURE OF THE WADGE HIERARCHY 121

61.3 Structure of the Wadge hierarchy

Martin and Monk proved in 1973 that AD implies the Wadge order for Baire space is well founded. Hence underAD, the Wadge classes modulo complements form a wellorder. The Wadge rank of a set A is the order type ofthe set of Wadge degrees modulo complements strictly below [ A ]W. The length of the Wadge hierarchy has beenshown to be Θ. Wadge also proved that the length of the Wadge hierarchy restricted to the Borel sets is φω1(1) (orφω1(2) depending on the notation), where φᵧ is the γth Veblen function to the base ω1 (instead of the usual ω).As for the Wadge lemma, this holds for any pointclass Γ, assuming the axiom of determinacy. If we associate witheach setA the collection of all sets strictly belowA on the Wadge hierarchy, this forms a pointclass. Equivalently, foreach ordinal α≤θ the collectionWα of sets which show up before stage α is a pointclass. Conversely, every pointclassis equal to someW α. A pointclass is said to be self-dual if it is closed under complementation. It can be shown thatWα is self-dual if and only if α is either 0, an even successor ordinal, or a limit ordinal of countable cofinality.

61.4 Other notions of degree

Similar notions of reduction and degree arise by replacing the continuous functions by any class of functions F whichcontains the identity function and is closed under composition. WriteA ≤FB ifA = f−1[B] for some function f inF. Any such class of functions again determines a preorder on the subsets of Baire space. Degrees given by Lipschitzfunctions are called Lipschitz degrees, and degrees from Borel functions Borel-Wadge degrees.

61.5 See also

• Analytical hierarchy

• Arithmetical hierarchy

• Axiom of determinacy

• Borel hierarchy

• Determinacy

• Pointclass

61.6 References

• Alexander S. Kechris; Benedikt Löwe; John R. Steel (eds.). Wadge Degrees and Projective Ordinals: The CabalSeminar Volume II. Lecture Notes in Logic. Cambridge University Press. ISBN 9781139504249.

• Andretta, Alessandro (2005). Bold, Stefan; Benedikt Löwe; Räsch, Thoralf et al., eds. “Infinite Games, Papersof the conference “Foundations of the Formal Sciences V” held in Bonn, Nov 26-29, 2004”. |chapter= ignored(help), in preparation

• Kanamori, Akihiro (2000). The Higher Infinite, second edition. Springer. ISBN 3-540-00384-3.

• Kechris, Alexander S. (1995). Classical Descriptive Set Theory. Springer. ISBN 0-387-94374-9.

• Wadge, William W. (1983). “Reducibility and determinateness on the Baire space”. PhD thesis. Univ. ofCalifornia, Berkeley.

61.7 Further reading

• Andretta, Alessandro and Martin, Donald (2003). “Borel-Wadge degrees”. Fundamenta Mathematicae 177(2): 175–192. doi:10.4064/fm177-2-5.

122 CHAPTER 61. WADGE HIERARCHY

• Cenzer, Douglas (1984). “Monotone Reducibility and the Family of Infinite Sets”. The Journal of SymbolicLogic (Association for Symbolic Logic) 49 (3): 774–782. doi:10.2307/2274130. JSTOR 2274130.

• Duparc, Jacques (2001). “Wadge hierarchy and Veblen hierarchy. Part I: Borel sets of finite rank”. Journal ofSymbolic Logic 66 (1): 55–86. doi:10.2307/2694911.

• Selivanov, Victor L. (2006). “Towards a descriptive set theory for domain-like structures”. Theoretical Com-puter Science Archive, Spatial representation: Discrete vs. Continuous computational models 365 (3): 258–282.doi:10.1016/j.tcs.2006.07.053. ISSN:0304-3975.

• Selivanov, Victor L. (2008). “Wadge Reducibility and Infinite Computations”. Mathematics in Computer Sci-ence 2 (1): 5–36. doi:10.1007/s11786-008-0042-x. ISSN:1661-8270.

• Semmes, Brian T. (2006). “A game for the Borel Functions”. preprint. Univ. of Amsterdam, ILLC Prepubli-cations PP-2006-24. Retrieved 2007-08-12.

Chapter 62

Zero-dimensional space

This article is about zero dimension in topology. For several kinds of zero space in algebra, see zero object (algebra).

In mathematics, a zero-dimensional topological space (or nildimensional) is a topological space that has dimensionzero with respect to one of several inequivalent notions of assigning a dimension to a given topological space.[1][2] Anillustration of a nildimensional space is a point.[3]

62.1 Definition

Specifically:

• A topological space is zero-dimensional with respect to the Lebesgue covering dimension if every finite opencover of the space has a finite refinement which is a cover of the space by open sets such that any point in thespace is contained in exactly one open set of this refinement.

• A topological space is zero-dimensional with respect to the small inductive dimension if it has a base consistingof clopen sets.

The two notions above agree for separable, metrisable spaces.

62.2 Properties of spaces with covering dimension zero• A zero-dimensional Hausdorff space is necessarily totally disconnected, but the converse fails. However, alocally compact Hausdorff space is zero-dimensional if and only if it is totally disconnected. (See (Arhangel’skii2008, Proposition 3.1.7, p.136) for the non-trivial direction.)

• Zero-dimensional Polish spaces are a particularly convenient setting for descriptive set theory. Examples ofsuch spaces include the Cantor space and Baire space.

• Hausdorff zero-dimensional spaces are precisely the subspaces of topological powers 2I where 2 = 0, 1 isgiven the discrete topology. Such a space is sometimes called a Cantor cube. If I is countably infinite, 2I isthe Cantor space.

62.3 Notes• Arhangel’skii, Alexander; Tkachenko, Mikhail (2008), Topological Groups and Related Structures, AtlantisStudies in Mathematics, Vol. 1, Atlantis Press, ISBN 90-78677-06-6

• Engelking, Ryszard (1977). General Topology. PWN, Warsaw.

• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

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124 CHAPTER 62. ZERO-DIMENSIONAL SPACE

62.4 References[1] “zero dimensional”. planetmath.org. Retrieved 2015-06-06.

[2] Hazewinkel, Michiel (1989). Encyclopaedia of Mathematics, Volume 3. Kluwer Academic Publishers. p. 190.

[3] Wolcott, Luke; McTernan, Elizabeth (2012). “ImaginingNegative-Dimensional Space” (PDF). In Bosch, Robert; McKenna,Douglas; Sarhangi, Reza. Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture. Phoenix, Arizona,USA: Tessellations Publishing. pp. 637–642. ISBN 978-1-938664-00-7. ISSN 1099-6702. Retrieved 10 July 2015.

Chapter 63

Θ (set theory)

In set theory, Θ (pronounced like the letter theta) is the least nonzero ordinal α such that there is no surjection fromthe reals onto α.If the axiom of choice (AC) holds (or even if the reals can be wellordered) then Θ is simply (2ℵ0)+ , the cardinalsuccessor of the cardinality of the continuum. However, Θ is often studied in contexts where the axiom of choicefails, such as models of the axiom of determinacy.Θ is also the supremum of the lengths of all prewellorderings of the reals.

63.1 Proof of existence

It may not be obvious that it can be proven, without using AC, that there even exists a nonzero ordinal onto whichthere is no surjection from the reals (if there is such an ordinal, then there must be a least one because the ordinalsare wellordered). However, suppose there were no such ordinal. Then to every ordinal α we could associate the setof all prewellorderings of the reals having length α. This would give an injection from the class of all ordinals intothe set of all sets of orderings on the reals (which can to be seen to be a set via repeated application of the powersetaxiom). Now the axiom of replacement shows that the class of all ordinals is in fact a set. But that is impossible, bythe Burali-Forti paradox.

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126 CHAPTER 63. Θ (SET THEORY)

63.2 Text and image sources, contributors, and licenses

63.2.1 Text• Adequate pointclass Source: https://en.wikipedia.org/wiki/Adequate_pointclass?oldid=590150799 Contributors: Ruud Koot, Epastore,

David Eppstein, R'n'B, J.delanoy, JackSchmidt, Hans Adler, Chimpionspeak and Anonymous: 1• Analytic set Source: https://en.wikipedia.org/wiki/Analytic_set?oldid=655921687 Contributors: AxelBoldt, Charles Matthews, Giftlite,

Crisófilax, Oleg Alexandrov, R.e.b., YurikBot, Trovatore, Deville, SmackBot, CBM, MystBot, Addbot, Citation bot, Citation bot 1 andChrisGualtieri

• Analytical hierarchy Source: https://en.wikipedia.org/wiki/Analytical_hierarchy?oldid=639041794Contributors: Michael Hardy, CharlesMatthews, Reddi, Dysprosia, Rich Farmbrough, Ben Standeven, EmilJ, MarSch, Hairy Dude, Trovatore, Deville, Arthur Rubin, Smack-Bot, JoeBot, Zero sharp, CBM, Gregbard, Cydebot, Pascal.Tesson, KudzuVine, El bot de la dieta, Addbot, Citation bot, Omnipaedistaand Anonymous: 3

• Arithmetical hierarchy Source: https://en.wikipedia.org/wiki/Arithmetical_hierarchy?oldid=679149133Contributors: AxelBoldt, Patrick,Michael Hardy, Bcrowell, Pde, Rotem Dan, Schneelocke, Charles Matthews, Wik, MathMartin, Kntg, Klemen Kocjancic, Alfeld, BenStandeven, Gauge, Kwamikagami, EmilJ, PeterMGerdes, Sligocki, Spambit, OlegAlexandrov, Fish-Face, NekoDaemon, Laubrau~enwiki,YurikBot, Hairy Dude, Gaius Cornelius, Trovatore, SmackBot, Maksim-e~enwiki, AlephNull~enwiki, Ligulembot, Tomlee2060, CR-Greathouse, CmdrObot, CBM, Gregbard, Cydebot, Pascal.Tesson, Knotwork, David Eppstein, LokiClock, PaulTanenbaum, Brezenbene,El bot de la dieta, Cnoguera, Addbot, Jarble, Yobot, Citation bot, VladimirReshetnikov, Kiefer.Wolfowitz, RobinK, Straightontillmorning,ZéroBot, ClueBot NG, Kükenschublade, Andsz, Subshift, Klilidiplomus, Deltahedron, Imperialfists, Roamingcuriosity and Anonymous:24

• Arithmetical set Source: https://en.wikipedia.org/wiki/Arithmetical_set?oldid=666692746 Contributors: Michael Hardy, Kku, Silver-fish, Hyacinth, MathMartin, Tobias Bergemann, Alexrexpvt, Peter M Gerdes, YurikBot, Trovatore, Maksim-e~enwiki, CBM, Cyde-bot, David Eppstein, R'n'B, Alexey Muranov, Addbot, VladimirReshetnikov, HRoestBot, ZéroBot, Rcsprinter123, ClueBot NG, Chris-Gualtieri, Knife-in-the-drawer and Anonymous: 3

• Axiom of projective determinacy Source: https://en.wikipedia.org/wiki/Axiom_of_projective_determinacy?oldid=606665075 Con-tributors: Charles Matthews, Ben Standeven, Gauge, Mairi, Trovatore, SmackBot, Bluebot, Ligulembot, Smith609, Hans Adler, DOI bot,Citation bot, Citation bot 1, BattyBot and Anonymous: 2

• Baire space (set theory) Source: https://en.wikipedia.org/wiki/Baire_space_(set_theory)?oldid=682015288 Contributors: TakuyaMu-rata, Revolver, Dcoetzee, Tobias Bergemann, Giftlite, Paul August, EmilJ, Salix alba, R.e.b., YurikBot, Wavelength, Trovatore, Lightcurrent, SmackBot, Bluebot, Ligulembot, CBM, Vadik, SieBot, Functor salad, Hccrle, Addbot, Yobot, Citation bot, Ravenousrepeller,WikitanvirBot, ZéroBot and Anonymous: 12

• Banach–Mazur game Source: https://en.wikipedia.org/wiki/Banach%E2%80%93Mazur_game?oldid=618515338 Contributors: Axel-Boldt, Michael Hardy, Revolver, Charles Matthews, Chtito, Oleg Alexandrov, Kzollman, Graham87, R.e.b., CiaPan, Algebraist, GaiusCornelius, Trovatore, Kewp, SmackBot, Selfworm, Stotr~enwiki, Kjs50, Magioladitis, Althai, PbBot, Skeptical scientist, Rtelgarsky,Addbot, Yobot, Howard McCay, ZéroBot, ChuispastonBot and Anonymous: 8

• Borel equivalence relation Source: https://en.wikipedia.org/wiki/Borel_equivalence_relation?oldid=634967192 Contributors: MichaelHardy, Charles Matthews, Tobias Bergemann, Paul August, Gauge, Ruud Koot, Trovatore, Bluebot, Ligulembot, Mets501, DOI bot,Yobot, Citation bot, Citation bot 1, Tkuvho, BattyBot, Brirush, Mark viking and Anonymous: 1

• Borel hierarchy Source: https://en.wikipedia.org/wiki/Borel_hierarchy?oldid=607157672 Contributors: Karada, Tobias Bergemann,Crisófilax, Touriste, Oleg Alexandrov, Jeff3000, Rjwilmsi, Algebraist, Trovatore, CBM, Just Chilling, Appraiser, ClueBot, Miaoku,Addbot, Yobot, Albertzeyer, Ripchip Bot, Jowa fan, שושן נר חיים and Anonymous: 3

• Borel right process Source: https://en.wikipedia.org/wiki/Borel_right_process?oldid=641626333 Contributors: Michael Hardy, Tompw,SmackBot, Whpq, Tdmg, Acs4b, Lfstevens, Magioladitis, Fabrictramp, Spartan-James, Nhleesdcawiki, Auntof6, Yobot, AnomieBOT,Feedintm and BattyBot

• Borel set Source: https://en.wikipedia.org/wiki/Borel_set?oldid=679386076 Contributors: AxelBoldt, Zundark, Miguel~enwiki, MichaelHardy, Isomorphic, TakuyaMurata, Ahoerstemeier, Revolver, Charles Matthews, Fibonacci, Sjorford, Romanm, Henrygb, Tobias Berge-mann, Weialawaga~enwiki, Giftlite, MathKnight, CSTAR, Vivacissamamente, Rich Farmbrough, Penis-breath, Zamfi, Harriv, Paul Au-gust, Bender235, Gauge, Rajah, Tsirel, BernardH, Cmapm, Oleg Alexandrov, Linas, Mike Segal, Margosbot~enwiki, YurikBot, Trovatore,DavidHouse~enwiki, Adpete, Nick Levine, Gala.martin, Lambiam, Jim.belk, Lilily, CBM, Harej bot, YK Times, Jay Gatsby, Albmont,Sullivan.t.j, Huzzlet the bot, Bongomatic, Policron, YohanN7, Quietbritishjim, SieBot, Stca74, Anchor Link Bot, CBM2, Denisarona, Es-tirabot, Alexey Muranov, Dthomsen8, Addbot, Delaszk, Yobot, Bdmy, Isheden, VladimirReshetnikov, Vectornaut, Yaddie, Xnn, Emaus-Bot, Rafi5749, Snorri, Y256, Limit-theorem, Mark viking and Anonymous: 34

• Cabal (set theory) Source: https://en.wikipedia.org/wiki/Cabal_(set_theory)?oldid=671232445 Contributors: AxelBoldt, Aleph4, D6,Rich Farmbrough, Mdd, Oleg Alexandrov, Rjwilmsi, Salix alba, Trovatore, SmackBot, Ligulembot, Gregbard, David Eppstein, Kope,Marekfull, Hans Adler, Citation bot, Monkbot and Anonymous: 2

• Cantor set Source: https://en.wikipedia.org/wiki/Cantor_set?oldid=685155764 Contributors: Damian Yerrick, AxelBoldt, Mav, Zun-dark, TheAnome, XJaM, Toby~enwiki, TobyBartels, Miguel~enwiki, Michael Hardy, Karada, Goatasaur, Schneelocke, CharlesMatthews,Dcoetzee, Nohat, Hyacinth, Pseudometric, AndrewKepert, Fibonacci, Marc Girod~enwiki, Robbot, Sverdrup, Choni, Bkell, Intangir,MOiRe, Aetheling, Ruakh, Dina, Tobias Bergemann, Tosha, Giftlite, Mshonle~enwiki, DavidCary, Fropuff, JeffBobFrank, Dmmaus,Mennucc, Fuzzy Logic, Sam Hocevar, Asbestos, Pyrop, TedPavlic, Gadykozma, Solkoll~enwiki, Marco Polo, Robotje, Townmouse,Eric Kvaalen, Complex01, Kotasik, Caesura, Derbeth, Lerdsuwa, Dzhim, Oleg Alexandrov, Hoziron, Simetrical, Linas, Plrk, Xaos-Bits, Cshirky, Mandarax, FlaBot, Mathbot, Xenobog, Fresheneesz, Chobot, YurikBot, RussBot, Zwobot, Tetracube, Vicarious, Jsnx,Selfworm, Neptunius, Chris the speller, Flyguy649, Kingdon, Daqu, Nishkid64, J. Finkelstein, Nicolas Bray, Loadmaster, Mets501,ILikeThings, CBM, Mattbuck, Reywas92, MC10, Robertinventor, Thijs!bot, Janviermichelle, Headbomb, Cj67, Salgueiro~enwiki, YKTimes, Beaumont, Magioladitis, Canter~enwiki, Cpiral, Policron, DavidCBryant, Adam1729, VolkovBot, TXiKiBoT, Cuinuc, Say some-thing then, Jesin, VanishedUserABC, Arcfrk, SieBot, Cwkmail, Swirlex, JackSchmidt, Blacklemon67, Readvanderbilt, Kompella, Ad-dbot, Mr.Xp, PV=nRT, ScAvenger, Soltanifar, Luckas-bot, Yobot, AnomieBOT, Materialscientist, DannyAsher, LilHelpa, Xqbot, Bdmy,

63.2. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 127

RJGray, GrouchoBot, RibotBOT, LucienBOT,DrilBot, Sierpinksis, Pjmcswee, Dlivnat, EmausBot, Tagib, Ginger Conspiracy, Vikram360,Hlange1, EdoBot, F. Weckenbarth, Delusion23, Mesoderm, Helpful Pixie Bot, Mohsen.soltanifar, Banausbal, Geilamir, Envictis, Blevin-tron, BlevintronBot, Dexbot, Brirush, Tina.baghaee, Nigellwh, K9re11, Owenengine, Mohsen-Soltanifar-UofT, Muztre and Anonymous:98

• Cantor space Source: https://en.wikipedia.org/wiki/Cantor_space?oldid=674578263 Contributors: AxelBoldt, Michael Hardy, Timwi,Saltine, MathMartin, Giftlite, GraemeBartlett, MarkusKrötzsch, Fropuff, Terrible Tim, DemonThing, Tomgally, OlegAlexandrov, Linas,Jclemens, R.e.b., Hairy Dude, Trovatore, Kompik, SmackBot, Melchoir, Bluebot, PieRRoMaN, Infovoria, CenozoicEra, Iridescent, CBM,Epbr123, David Eppstein, Enviroboy, Tresiden, ClueBot, Yasmar, Cenarium, Luca Antonelli, Addbot, Jewelz 17, Yobot, AnomieBOT,Citation bot, HiW-Bot, D.Lazard, Solomon7968, Muztre and Anonymous: 19

• Choquet game Source: https://en.wikipedia.org/wiki/Choquet_game?oldid=616713034 Contributors: Michael Hardy and R.e.b.• Cichoń's diagram Source: https://en.wikipedia.org/wiki/Cicho%C5%84’s_diagram?oldid=609130049Contributors: Zundark, Charles

Matthews, Aleph4, Ben Standeven, Gene Nygaard, Dtrebbien, JRSpriggs, Ntsimp, Headbomb, David Eppstein, Turtlens, Andrewbt,Rumping, Addbot, Yobot, Citation bot, Citation bot 1, Helpful Pixie Bot, Deltahedron and Anonymous: 5

• Coanalytic set Source: https://en.wikipedia.org/wiki/Coanalytic_set?oldid=513879204 Contributors: Michael Hardy, Blotwell, Salixalba, Trovatore, SmackBot, TheKMan, Sixfooteskimo, CBM, Hamish Ross!!, Magioladitis, Vadik, Ouedbirdwatcher, Hans Adler, Clue-Bot NG and Anonymous: 1

• Descriptive set theory Source: https://en.wikipedia.org/wiki/Descriptive_set_theory?oldid=674751154 Contributors: Revolver, CharlesMatthews, Dysprosia, Maximus Rex, Tosha, Giftlite, Blotwell, Msh210, Trovatore, SmackBot, Gala.martin, Stotr~enwiki, CBM, Pas-cal.Tesson, Thijs!bot, WinBot, Perelaar, JohnBlackburne, Mrw7, Henry Delforn (old), JerroldPease-Atlanta, Addbot, LaaknorBot, Yobot,Citation bot, Tkuvho, Onel5969, ZéroBot, Brad7777, Mark viking and Anonymous: 15

• Difference hierarchy Source: https://en.wikipedia.org/wiki/Difference_hierarchy?oldid=649005685Contributors: Zundark, EmilJ,MarSch,Coremodel, CBM, Pascal.Tesson, David Eppstein, Hans Adler, Yobot, Omnipaedista and Erik9bot

• Effective descriptive set theory Source: https://en.wikipedia.org/wiki/Effective_descriptive_set_theory?oldid=643982249 Contribu-tors: Fbkintanar, Salix alba, Trovatore, That Guy, From That Show!, SmackBot, CBM, Hans Adler, Citation bot, George Pelltier, Brirushand Anonymous: 2

• Effective Polish space Source: https://en.wikipedia.org/wiki/Effective_Polish_space?oldid=626071484 Contributors: Tobias Berge-mann, CBM, Cydebot, RebelRobot, CBM2, Andrewbt and Helpful Pixie Bot

• Fσ set Source: https://en.wikipedia.org/wiki/F%CF%83_set?oldid=612010054 Contributors: Michael Hardy, Karada, Revolver, Dcoet-zee, Dysprosia, MathMartin, Fropuff, Paul August, RJFJR, WojciechSwiderski~enwiki, Oleg Alexandrov, Marudubshinki, Bgwhite,Epolk, Trovatore, Melchoir, Nbarth, Ravi12346, CBM,Michael Fourman, David Eppstein, Smithers888, Remember the dot, JackSchmidt,Addbot, AnomieBOT, Erik9bot, DrilBot, Grzjnck, Vivagra, ZéroBot, Savick01 and Anonymous: 4

• Gregory number Source: https://en.wikipedia.org/wiki/Gregory_number?oldid=618841543Contributors: XJaM,Michael Hardy, Prime-Fan, Giftlite, PrimeHunter, David Eppstein, Bikasuishin, Addbot, Monkbot and Anonymous: 2

• Gδ set Source: https://en.wikipedia.org/wiki/G%CE%B4_set?oldid=660190370 Contributors: The Anome, Patrick, Michael Hardy, Re-volver, Dcoetzee, Dysprosia, Kevinatilusa, MathMartin, Bkell, Fropuff, Paul August, Bender235, Tsirel, WojciechSwiderski~enwiki,Oleg Alexandrov, Mathbot, Bgwhite, Algebraist, Trovatore, Melchoir, Nbarth, Makyen, Mets501, CBM, Michael Fourman, Vanish2,Hans Lundmark, David Eppstein, Remember the dot, PMajer, Plclark, Arcfrk, YohanN7, JackSchmidt, Blacklemon67, Jaan Vajakas,Addbot, Topology Expert, Zorrobot, Yobot, FrescoBot, Citation bot 1, DrilBot, EmausBot, ZéroBot, ClueBot NG, Sopasakis p, BG19bot,CitationCleanerBot, K9re11, Monkbot and Anonymous: 14

• Homogeneous tree Source: https://en.wikipedia.org/wiki/Homogeneous_tree?oldid=605828305 Contributors: Mrwright, Hans Adler,DOI bot, Citation bot 1 and BattyBot

• Homogeneously Suslin set Source: https://en.wikipedia.org/wiki/Homogeneously_Suslin_set?oldid=605828312 Contributors: Dman-ning, Mrwright, SmackBot, McPoet, Zrustin, Hans Adler, DOI bot, Citation bot 1 and BattyBot

• Inductive set Source: https://en.wikipedia.org/wiki/Inductive_set?oldid=607144947Contributors: TakuyaMurata, Trovatore, SmackBot,PieRRoMaN, Ligulembot, Hans Adler, Yobot, Citation bot, Kephir and Anonymous: 3

• Infinity-Borel set Source: https://en.wikipedia.org/wiki/Infinity-Borel_set?oldid=607145082 Contributors: Zundark, Schneelocke, To-bias Bergemann, Ben Standeven, Oleg Alexandrov, Mathbot, Joth, Trovatore, SmackBot, Melchoir, Yobot, Erik9bot and Chimpionspeak

• Interval (mathematics) Source: https://en.wikipedia.org/wiki/Interval_(mathematics)?oldid=684369036 Contributors: Zundark, EdPoor, JeLuF, Patrick, Michael Hardy, Andres, Bjcairns, Charles Matthews, Dcoetzee, Dysprosia, Jitse Niesen, OlivierM, McKay, Robbot,Ruinia, Ojigiri~enwiki, Ambarish, Tobias Bergemann, Alan Liefting, Tosha, Giftlite, Markus Krötzsch, MSGJ, Markus Kuhn, GarethWyn, Jorge Stolfi, Gazibara, Sam Hocevar, PhotoBox, Mormegil, Paul August, Rgdboer, Liberatus, EmilJ, Jpgordon, Teorth, Jsoulie,Jumbuck, Burn, Krellion, Oleg Alexandrov, Simetrical, Mindmatrix, Ikescs, OdedSchramm, Rejnal, Theboywonder, Dpr, Dpv, Salixalba, Mathbot, Greg321, AttishOculus, Gurch, Dmitry-kazakov, Chobot, YurikBot, Hairy Dude, RussBot, Michael Slone, Stephenb,Gaius Cornelius, PaulGarner, Scilicet, GrinBot~enwiki, JJL, SmackBot, Incnis Mrsi, Tomáš Petříček, SmartGuy Old, Silly rabbit, Nbarth,DHN-bot~enwiki, Cybercobra, SashatoBot, Cronholm144, Amine Brikci N, Pet-ro, Eassin, Ylloh, CRGreathouse, Woudloper, CBM, HeWho Is, Thijs!bot, Epbr123, Jojan, Pjvpjv, Marek69, NERIUM, Opelio, JAnDbot, Martinkunev, Nyq, Albmont, Email4mobile, DavidEppstein, Anaxial, J.delanoy, Pharaoh of the Wizards, Extransit, Redrad, NewEnglandYankee, Cometstyles, DavidCBryant, Hulten, The-NewPhobia, Pleasantville, Philip Trueman, TXiKiBoT, Enviroboy, Insanity Incarnate, Dmcq, Monty845, Cowlinator, Quietbritishjim,SieBot, ToePeu.bot, Nathan B. Kitchen, Ezh, Lightmouse, Skeptical scientist, Msrasnw, Anchor Link Bot, ClueBot, Rumping, Marino-slo,The Thing That Should Not Be, Idleloop~enwiki, CptCutLess, Otolemur crassicaudatus, Excirial, Bremerenator, Hans Adler, Schreiber-Bike, Ottawa4ever, Kikos, SoxBot III, Knopfkind, SilvonenBot, Addbot, Mammothx, LaaknorBot, Glane23, AndersBot, DavidBParker,Jasper Deng, 5 albert square, Tide rolls, Zorrobot, Luckas-bot, Yobot, Vs64vs, JorgeFierro, Allent511, AnomieBOT, Qdinar, AdjustShift,RandomAct, Flewis, Xelnx, ArthurBot, Pownuk, LilHelpa, Obersachsebot, Capricorn42, SteveWoolf, TonyHagale, Charvest, Aghajan-pour, Shadowjams, LucienBOT, Calmer Waters, ItsZippy, Sumone10154, Jurryaany, DARTH SIDIOUS 2, Woogee, Hyarmendacil,Noommos, Jowa fan, EmausBot, K6ka, John Cline, Quondum, Wayne Slam, Sassospicco, Mayur, Wikiloop, Bean49Bot, DASHBotAV,28bot, ClueBot NG, Jack Greenmaven, Wcherowi, Tideflat, Widr, Vibhijain, Helpful Pixie Bot, ,اقرأ Webclient101, Stephan Kulla, I amOne of Many, Theopolito, Ginsuloft, KH-1 and Anonymous: 205

128 CHAPTER 63. Θ (SET THEORY)

• Kleene–Brouwer order Source: https://en.wikipedia.org/wiki/Kleene%E2%80%93Brouwer_order?oldid=551286561Contributors: CharlesMatthews, Rich Farmbrough, Manta~enwiki, Trovatore, Mrwright, Cronholm144, David Eppstein, Yobot, Erik9bot, Wgunther, Fairflow,ChrisGualtieri and Anonymous: 2

• Kuratowski–Ulam theorem Source: https://en.wikipedia.org/wiki/Kuratowski%E2%80%93Ulam_theorem?oldid=660612920Contrib-utors: Michael Hardy, Piotrus, Bender235, R.e.b., BG19bot and K9re11

• L(R) Source: https://en.wikipedia.org/wiki/L(R)?oldid=635375890 Contributors: Zundark, Schneelocke, Giftlite, Waltpohl, Mysidia,Ben Standeven, Trovatore, Pgk, Bluebot, Ligulembot, Beetstra, JRSpriggs, DOI bot, Citation bot 1, Brirush and Anonymous: 2

• Lightface analytic game Source: https://en.wikipedia.org/wiki/Lightface_analytic_game?oldid=631352163 Contributors: Sgeo, Trova-tore, TechnoGuyRob, CBM, AnomieBOT, Erik9bot and Mark viking

• List of properties of sets of reals Source: https://en.wikipedia.org/wiki/List_of_properties_of_sets_of_reals?oldid=607058217 Con-tributors: Charles Matthews, ZeroOne, Oleg Alexandrov, Salix alba, Mathbot, Trovatore, Syrcatbot, Yoni and Skeptical scientist

• Lusin’s separation theorem Source: https://en.wikipedia.org/wiki/Lusin’s_separation_theorem?oldid=616041706Contributors: MichaelHardy, Giftlite, R.e.b., Sodin, CBM, Magioladitis, Yoni, Daniele.tampieri, Helpful Pixie Bot and Anonymous: 1

• Luzin space Source: https://en.wikipedia.org/wiki/Luzin_space?oldid=624811711 Contributors: Charles Matthews, R.e.b., Trovatore,Incnis Mrsi, Via strass, Addbot, Kilom691, Citation bot and Anonymous: 1

• Martin measure Source: https://en.wikipedia.org/wiki/Martin_measure?oldid=657136961 Contributors: Oleg Alexandrov, Rjwilmsi,Mrwright, GiantSnowman, CBM, Cydebot, David Eppstein and FrescoBot

• Meagre set Source: https://en.wikipedia.org/wiki/Meagre_set?oldid=660613711 Contributors: Toby Bartels, Charles Matthews, To-bias Bergemann, Giftlite, ReiVaX, Paul August, Bender235, Clement Cherlin, Touriste, Eric Kvaalen, YurikBot, Trovatore, Smack-Bot, Nbarth, JRSpriggs, CBM, Vnoort, Goldencako, Salgueiro~enwiki, Magioladitis, Lucianoidecastro, TXiKiBoT, Skeptical scientist,UKoch, Alexbot, Addbot, DOI bot, Luckas-bot, RibotBOT, Howard McCay, Frietjes, MaximalIdeal, Mark viking, Mgkrupa and Anony-mous: 13

• Negative-dimensional space Source: https://en.wikipedia.org/wiki/Negative-dimensional_space?oldid=673554250Contributors: MichaelHardy, Ogerard, Lamro, Northernhenge and Anonymous: 1

• Nested intervals Source: https://en.wikipedia.org/wiki/Nested_intervals?oldid=631120148 Contributors: Zundark, Charles Matthews,Giftlite, EmilJ, HannsEwald, SmackBot, Hydrogen Iodide, JCSantos, Radagast83, Jim.belk, Vanisaac, David Eppstein, Myrkkyhammas,VolkovBot, Geometry guy, Le Pied-bot~enwiki, Thehotelambush, Addbot, Erik9bot, ZéroBot, ChuispastonBot, Brad7777, DeathOfBal-ance, SeriouslySmart and Anonymous: 6

• Normal number Source: https://en.wikipedia.org/wiki/Normal_number?oldid=661159750Contributors: AxelBoldt, TheAnome, XJaM,Patrick, Michael Hardy, GTBacchus, Charles Matthews, Timwi, Jitse Niesen, Fredrik, Tualha, Bkell, Robinh, HaeB, Mfc, Giftlite, Elf,Smurfix, Anton Mravcek, Golbez, Peter Kwok, Mschlindwein, Finog, Mike Rosoft, Jnestorius, El C, Flammifer, Haham hanuka, Hashar-Bot~enwiki, Mjpotter, Joriki, Isnow, Rjwilmsi, Fieari, JVz, Pexatus, Don Gosiewski, Nmondal, Trovatore, R.e.s., Expensivehat, DYLANLENNON~enwiki, Arthur Rubin, Vogelfrei, A Doon, SmackBot, Incnis Mrsi, Melchoir, Bluebot, Zgyorfi~enwiki, Lambiam, Loadmas-ter, Stephen B Streater, Amakuru, CRGreathouse, CmdrObot, CBM, Myasuda, Stormwyrm, Mon4, Optimist on the run, Xantharius,Thijs!bot, Oerjan, Headbomb, Paxinum, Davidhorman, Flarity, Ste4k, Hermel, Dricherby, Magioladitis, VoABot II, Robomojo, DavidEppstein, Kevinsam, Fylwind, VolkovBot, Davidwr, Yugsdrawkcabeht, Dojarca, GroveGuy, Lamro, Wpegden, Tlepp, Addbot, DOI bot,Luckas-bot, Mon oncle, AnomieBOT, Citation bot, ArthurBot, FrescoBot, Citation bot 1, RjwilmsiBot, Set theorist, Helpful Pixie Bot,Deltahedron, Spectral sequence, Jochen Burghardt, Andyhowlett, Babak.k.shandiz, Ben István and Anonymous: 59

• Perfect set property Source: https://en.wikipedia.org/wiki/Perfect_set_property?oldid=488354535 Contributors: Gauge, Mathbot, Al-gebraist, Trovatore, SmackBot, CBM and Anonymous: 5

• Pointclass Source: https://en.wikipedia.org/wiki/Pointclass?oldid=680519587Contributors: MathMartin, Salix alba,MZMcBride,Wave-length, Trovatore, SmackBot, Zero sharp, CBM, Reedy Bot, The enemies of god, Yobot and The Quixotic Potato

• Polish group Source: https://en.wikipedia.org/wiki/Polish_group?oldid=625336781 Contributors: R.e.b., Macdstu and Jumpythehat• Polish space Source: https://en.wikipedia.org/wiki/Polish_space?oldid=681814618 Contributors: Michael Hardy, TakuyaMurata, An-

gela, Revolver, Charles Matthews, Tosha, Gauge, EmilJ, Oleg Alexandrov, Linas, Fbkintanar, MarSch, R.e.b., Trovatore, Kompik, ThatGuy, From That Show!, Gala.martin, Mathsci, Stotr~enwiki, Easwaran, Zero sharp, A. Pichler, CBM, RogierBrussee, Sullivan.t.j, Dave-crosby uk, Sapphic, Nihil novi, Pernambuko, Addbot, Yobot, Hairer, AnomieBOT, Citation bot, Xqbot, Macdstu, Omnipaedista, Flurm-flam, HighCrossRuff, WikitanvirBot, Slawekb, MaximalIdeal, Cleanelephant, Mark viking, Julian.cardich, Mgkrupa, Teddyktchan andAnonymous: 21

• Prewellordering Source: https://en.wikipedia.org/wiki/Prewellordering?oldid=488790287Contributors: Zundark, Patrick, CharlesMatthews,EmilJ, Oleg Alexandrov, NickBush24, Trovatore, Tetracube, That Guy, From That Show!, SmackBot, Nbarth, Henry Delforn (old), Pal-not, Citation bot and Anonymous: 3

• Projective hierarchy Source: https://en.wikipedia.org/wiki/Projective_hierarchy?oldid=650939657 Contributors: AxelBoldt, Zundark,Charles Matthews, AshtonBenson, Wtmitchell, Trovatore, Stotr~enwiki, CBM, R'n'B, Addbot, ZéroBot and Anonymous: 2

• Property of Baire Source: https://en.wikipedia.org/wiki/Property_of_Baire?oldid=625995113Contributors: TakuyaMurata, JitseNiesen,Fibonacci, Aleph4, Woohookitty, Manta~enwiki, Trovatore, Mr Stephen, Stotr~enwiki, Zero sharp, Road Wizard, David Eppstein, Ad-dbot, Luckas-bot, ZéroBot, Helpful Pixie Bot, BG19bot, Mark viking and Anonymous: 3

• Scale (descriptive set theory) Source: https://en.wikipedia.org/wiki/Scale_(descriptive_set_theory)?oldid=586972236Contributors: Mdd,Trovatore, SmackBot, JL-Bot, ClueBot, Hans Adler, Materialscientist, BG19bot, 8598gratee, ChrisGualtieri and Anonymous: 2

• Schröder–Bernstein theorem formeasurable spaces Source: https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem_for_measurable_spaces?oldid=650609168 Contributors: Zundark, Michael Hardy, Giftlite, Rich Farmbrough, Tsirel, Sodin,RDBury, BillFlis, Rschwieb, Blehfu, Good Olfactory, Yobot, Omnipaedista, Brad7777 and Jochen Burghardt

• Smith–Volterra–Cantor set Source: https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set?oldid=684863115Contributors: Michael Hardy, Charles Matthews, Henrygb, Tobias Bergemann, Tosha, Lethe, Squash, Gauge, Tsirel, Linas, Henry Bottom-ley, Fresheneesz, Tetracube, AndrewWTaylor, Daqu, Mr Death, Mets501, Colin Rowat, David Eppstein, Ernest lk lam, Addbot, Uncia,Yobot, ArthurBot, Xqbot, GrouchoBot, LucienBOT, Thomassteinke and Anonymous: 12

63.2. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 129

• Stonehamnumber Source: https://en.wikipedia.org/wiki/Stoneham_number?oldid=607170457Contributors: Schneelocke, CharlesMatthews,Oleg Alexandrov, Linas, R.e.s., Catapult, CRGreathouse, David Eppstein, DumZiBoT, Addbot, LaaknorBot, Lightbot, Yobot, Rjwilmsi-Bot, Deltahedron and Anonymous: 2

• Suslin operation Source: https://en.wikipedia.org/wiki/Suslin_operation?oldid=621866051 Contributors: R.e.b., Pmckenney and Lupin-noble90

• Tree (descriptive set theory) Source: https://en.wikipedia.org/wiki/Tree_(descriptive_set_theory)?oldid=614358466Contributors: CharlesMatthews, GPHemsley, Aleph4, Andreas Kaufmann, Rich Farmbrough, Diego Moya, R.e.b., Trovatore, Kompik, Abstract Idiot, DavidEppstein, Yobot, AnomieBOT, LilHelpa, False vacuum, Erik9bot, ChrisGualtieri, Eduenez and Anonymous: 1

• Uniformization (set theory) Source: https://en.wikipedia.org/wiki/Uniformization_(set_theory)?oldid=526843778Contributors: MichaelHardy, Schneelocke, Aleph4, Dbenbenn, Paul August, Oleg Alexandrov, R.e.b., Trovatore, That Guy, From That Show!, SmackBot, JR-Spriggs, Citation bot and Anonymous: 1

• Unit interval Source: https://en.wikipedia.org/wiki/Unit_interval?oldid=667649769Contributors: AxelBoldt, TimoHonkasalo, Toby~enwiki,Michael Hardy, Charles Matthews, Jitse Niesen, Hyacinth, Francs2000, Tobias Bergemann, Giftlite, Fropuff, Sam nead, Rgdboer, ABCD,Jacobolus, Teknic, Melchoir, Silly rabbit, Nbarth, UU, CRGreathouse, CBM, Cydebot, Al Lemos, VictorAnyakin, Vanish2, Cpiral,VolkovBot, Jamelan, Dogah, Paolo.dL, Melcombe, Qwfp, MystBot, Addbot, Topology Expert, Luckas-bot, LilHelpa, GrouchoBot, Xnn,EmausBot, Jaydiem and Anonymous: 9

• Universally Baire set Source: https://en.wikipedia.org/wiki/Universally_Baire_set?oldid=491706639 Contributors: Rjwilmsi, Trovatore,David Eppstein, Hans Adler and Helpful Pixie Bot

• Universally measurable set Source: https://en.wikipedia.org/wiki/Universally_measurable_set?oldid=607166859 Contributors: CharlesMatthews, Oleg Alexandrov, Mathbot, Trovatore, Sandstein, CBM, Yobot and Helpful Pixie Bot

• Vitali set Source: https://en.wikipedia.org/wiki/Vitali_set?oldid=675241760Contributors: Michael Hardy, Loisel, Cyan, Revolver, CharlesMatthews, Dino, Fibonacci, Bkell, UtherSRG,Aetheling, JerryFriedman, Tobias Bergemann, Tosha, Giftlite, IanMaxwell, SimonLacoste-Julien, Moxfyre, Vivacissamamente, Gadykozma, ArnoldReinhold, Crisófilax, Touriste, Army1987, BernardH, Oleg Alexandrov, Linas,Rictus, Hgkamath, R.e.b., Chobot, YurikBot, Archelon, AlexeiK, Petter Strandmark, Trovatore, RDBury, Karl Stroetmann, Lim WeiQuan, TooMuchMath, W3asal, Salgueiro~enwiki, JJ Harrison, Sullivan.t.j, Friday529, Polkaparty, PaulTanenbaum, Likebox, MaSt, Ad-dbot, PV=nRT, Luckas-bot, Yobot, LGB, Piano non troppo, Constructive editor, Mafaraxas, Conjugado, ZéroBot, WeijiBaikeBianji,BG19bot, Max Longint, Makecat-bot, Zamir1234, Yair-koren-1977 and Anonymous: 36

• Wadge hierarchy Source: https://en.wikipedia.org/wiki/Wadge_hierarchy?oldid=682348434 Contributors: Michael Hardy, Karada, BenStandeven, Walkiped, Mdd, Spambit, Rjwilmsi, BradBeattie, Trovatore, SmackBot, Melchoir, Mirasmus, Nbarth, Mets501, Iridescent,CBM, Ruthwdg, Ensign beedrill, JaGa, S, Davidwr, M gol, Miaoku, DOI bot, Atonkf, Citation bot, FrescoBot, Citation bot 1, Jonesey95,Mehmetorgun, Solomon7968 and Anonymous: 16

• Zero-dimensional space Source: https://en.wikipedia.org/wiki/Zero-dimensional_space?oldid=676641292 Contributors: The Anome,Dominus, Tobias Bergemann, Mporter, Paul August, Vipul, Linas, YurikBot, Trovatore, Kompik, Arthur Rubin, That Guy, From ThatShow!, SmackBot, Incnis Mrsi, Melchoir, Vina-iwbot~enwiki, Cesium 133, Stotr~enwiki, P2005t, Dp462090, Cydebot, Ntsimp, R'n'B,Squad51, Trumpet marietta 45750, Anonymous Dissident, Plclark, Lamro, SieBot, Mojoworker, Mitch Ames, Addbot, Numbo3-bot,Luckas-bot, Yobot, 4th-otaku, Citation bot, IVAN3MAN, Dr. John D. McCarthy, Drusus 0, D.Lazard, ClueBot NG, Helpful Pixie Bot,ChrisGualtieri, Mark viking, Jjbernardiscool, Noyster and Anonymous: 10

• Θ(set theory) Source: https://en.wikipedia.org/wiki/%CE%98_(set_theory)?oldid=601145089Contributors: Zundark, Jerzy, OneWeird-Dude, Bgwhite, Trovatore, Calliopejen~enwiki, CBM, David Eppstein, Ddd1600, Hans Adler, Yobot, AnomieBOT, Erik9bot, JumpDis-cont and Anonymous: 2

63.2.2 Images• File:Ambox_important.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg License: Public do-

main Contributors: Own work, based off of Image:Ambox scales.svg Original artist: Dsmurat (talk · contribs)• File:Cantor_dust.png Source: https://upload.wikimedia.org/wikipedia/commons/6/65/Cantor_dust.png License: Copyrighted free use

Contributors: http://en.wikipedia.org/wiki/Image:Cantor_dust.png Original artist: Solkoll• File:Cantor_set_in_seven_iterations.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/56/Cantor_set_in_seven_iterations.

svg License: Public domain Contributors: From en.wikipedia.org Image:Cantor_set_in_seven_iterations.svg Original artist: 127 “rect” <ahref='//commons.wikimedia.org/wiki/File:W3C_valid.svg' class='image' title='This image is valid SVG'><img alt='This image is validSVG' src='https://upload.wikimedia.org/wikipedia/commons/thumb/6/66/W3C_valid.svg/32px-W3C_valid.svg.png' width='32' height='16'srcset='https://upload.wikimedia.org/wikipedia/commons/thumb/6/66/W3C_valid.svg/48px-W3C_valid.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/6/66/W3C_valid.svg/64px-W3C_valid.svg.png 2x' data-file-width='200' data-file-height='100' /></a>

• File:Cantors_cube.jpg Source: https://upload.wikimedia.org/wikipedia/commons/7/78/Cantors_cube.jpg License: Public domain Con-tributors:

• en:Image:Cantors cube.jpg Original artist: User:Solkoll• File:CardContin.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/75/CardContin.svg License: Public domain Contrib-

utors: en:Image:CardContin.png Original artist: en:User:Trovatore, recreated by User:Stannered• File:E-to-the-i-pi.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/35/E-to-the-i-pi.svg License: CC BY 2.5 Contribu-

tors: ? Original artist: ?• File:Fractal_fern_explained.png Source: https://upload.wikimedia.org/wikipedia/commons/4/4b/Fractal_fern_explained.pngLicense:

Public domain Contributors: Own work Original artist: António Miguel de Campos• File:Illustration_nested_intervals.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/48/Illustration_nested_intervals.svg

License: CC BY-SA 3.0 Contributors: Own work Original artist: Stephan Kulla (User:Stephan Kulla)

130 CHAPTER 63. Θ (SET THEORY)

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