Ordinal Number 8

Ordinal numberFrom Wikipedia, the free encyclopedia

Contents

1 Bijection 11.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Batting line-up of a baseball team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Seats and students of a classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 More mathematical examples and some non-examples . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Bijections and cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 Bijections and category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.9 Generalization to partial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.10 Contrast with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Cardinal number 72.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Cardinal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.1 Successor cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.2 Cardinal addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.3 Cardinal multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.4 Cardinal exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 The continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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3 Cardinality 153.1 Comparing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Denition 1: | A | = | B | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.2 Denition 2: | A | | B | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.3 Denition 3: | A | < | B | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Cardinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Finite, countable and uncountable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Innite sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4.1 Cardinality of the continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6 Union and intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Coniteness 204.1 Boolean algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Conite topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2.2 Double-pointed conite topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3.1 Product topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3.2 Direct sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Derived set (mathematics) 225.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Topology in terms of derived sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 CantorBendixson rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Discrete space 246.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.4 Indiscrete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.5 Quotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7 First uncountable ordinal 277.1 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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7.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

8 Georg Cantor 288.1 Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

8.1.1 Youth and studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.1.2 Teacher and researcher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.1.3 Late years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8.2 Mathematical work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.2.1 Number theory, trigonometric series and ordinals . . . . . . . . . . . . . . . . . . . . . . 318.2.2 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

8.3 Philosophy, religion and Cantors mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.4 Cantors ancestry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.5 Historiography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9 Hereditary property 469.1 In topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.2 In graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

9.2.1 Monotone property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.3 In model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.4 In matroid theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.5 In set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

10 Injective function 4910.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.3 Injections can be undone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410.4 Injections may be made invertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410.5 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410.6 Proving that functions are injective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

11 Integer 5711.1 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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11.2 Order-theoretic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.3 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.4 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6011.5 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6011.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6111.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6111.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6111.9 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6211.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

12 Natural number 6312.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

12.1.1 Modern denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6512.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6512.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

12.3.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6512.3.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.3.3 Relationship between addition and multiplication . . . . . . . . . . . . . . . . . . . . . . . 6612.3.4 Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.3.5 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.3.6 Algebraic properties satised by the natural numbers . . . . . . . . . . . . . . . . . . . . . 66

12.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6712.5 Formal denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

12.5.1 Peano axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6712.5.2 Constructions based on set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

12.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6912.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6912.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7012.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

13 Order isomorphism 7413.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7413.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7413.3 Order types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7413.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

14 Order type 7614.1 Order type of well-orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7614.2 Rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

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14.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

15 Ordinal arithmetic 7815.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7815.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.3 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8015.4 Cantor normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8215.5 Factorization into primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8315.6 Large countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8315.7 Natural operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8415.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8515.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8515.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

16 Ordinal number 8616.1 Ordinals extend the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8716.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

16.2.1 Well-ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8916.2.2 Denition of an ordinal as an equivalence class . . . . . . . . . . . . . . . . . . . . . . . 8916.2.3 Von Neumann denition of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8916.2.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

16.3 Transnite sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9016.4 Transnite induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

16.4.1 What is transnite induction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9016.4.2 Transnite recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9116.4.3 Successor and limit ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9116.4.4 Indexing classes of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9116.4.5 Closed unbounded sets and classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

16.5 Arithmetic of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9216.6 Ordinals and cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

16.6.1 Initial ordinal of a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9316.6.2 Conality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

16.7 Some large countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9316.8 Topology and ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9416.9 Downward closed sets of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9416.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9416.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9416.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9416.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

vi CONTENTS

17 Partially ordered set 9617.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9717.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9717.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9717.4 Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . 9817.5 Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9817.6 Strict and non-strict partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9917.7 Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9917.8 Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9917.9 Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10017.10Linear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10017.11In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10117.12Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10117.13Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10117.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10117.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10217.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10217.17External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

18 Set theory 10318.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10418.2 Basic concepts and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10518.3 Some ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10618.4 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10618.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10718.6 Areas of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

18.6.1 Combinatorial set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10818.6.2 Descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10818.6.3 Fuzzy set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10818.6.4 Inner model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10818.6.5 Large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10918.6.6 Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10918.6.7 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10918.6.8 Cardinal invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10918.6.9 Set-theoretic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

18.7 Objections to set theory as a foundation for mathematics . . . . . . . . . . . . . . . . . . . . . . . 11018.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11018.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11018.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11118.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

19 Surjective function 112

CONTENTS vii

19.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11319.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11319.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

19.3.1 Surjections as right invertible functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11419.3.2 Surjections as epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11519.3.3 Surjections as binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11619.3.4 Cardinality of the domain of a surjection . . . . . . . . . . . . . . . . . . . . . . . . . . . 11619.3.5 Composition and decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11619.3.6 Induced surjection and induced bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

19.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11619.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11719.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

20 Topological space 11820.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

20.1.1 Neighbourhoods denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11820.1.2 Open sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11920.1.3 Closed sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12020.1.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

20.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12020.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12020.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12120.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12220.6 Classication of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12220.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12220.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12220.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12220.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12320.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12320.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12320.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

21 Total order 12521.1 Strict total order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12521.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12621.3 Further concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

21.3.1 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12621.3.2 Lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12621.3.3 Finite total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12721.3.4 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12721.3.5 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12721.3.6 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

viii CONTENTS

21.3.7 Sums of orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12721.4 Orders on the Cartesian product of totally ordered sets . . . . . . . . . . . . . . . . . . . . . . . . 12821.5 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12821.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12821.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12821.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

22 Transitive set 13022.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13022.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13022.3 Transitive closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13022.4 Transitive models of set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13022.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13122.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13122.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

23 Trigonometric series 13223.1 The zeros of a trigonometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13223.2 Zygmunds book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13223.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

23.3.1 Reviews of Trigonometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13323.3.2 Publication history of Trigonometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . 133

24 Von Neumann cardinal assignment 13424.1 Initial ordinal of a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13424.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13424.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

25 Well-order 13625.1 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13625.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

25.2.1 Natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13725.2.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13725.2.3 Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

25.3 Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13825.4 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13825.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13925.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13925.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 140

25.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14025.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14525.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Chapter 1

Bijection

X 1

2

3

4

YD

B

C

A

A bijective function, f: X Y, where set X is {1, 2, 3, 4} and set Y is {A, B, C, D}. For example, f(1) = D.

In mathematics, a bijection, bijective function or one-to-one correspondence is a function between the elementsof two sets, where every element of one set is paired with exactly one element of the other set, and every elementof the other set is paired with exactly one element of the rst set. There are no unpaired elements. In mathematical

1

2 CHAPTER 1. BIJECTION

terms, a bijective function f: X Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.A bijection from the setX to the set Y has an inverse function from Y toX. IfX and Y are nite sets, then the existenceof a bijection means they have the same number of elements. For innite sets the picture is more complicated, leadingto the concept of cardinal number, a way to distinguish the various sizes of innite sets.A bijective function from a set to itself is also called a permutation.Bijective functions are essential tomany areas ofmathematics including the denitions of isomorphism, homeomorphism,dieomorphism, permutation group, and projective map.

1.1 DenitionFor more details on notation, see Function (mathematics) Notation.

For a pairing between X and Y (where Y need not be dierent from X) to be a bijection, four properties must hold:

1. each element of X must be paired with at least one element of Y,

2. no element of X may be paired with more than one element of Y,

3. each element of Y must be paired with at least one element of X, and

4. no element of Y may be paired with more than one element of X.

Satisfying properties (1) and (2) means that a bijection is a function with domain X. It is more common to seeproperties (1) and (2) written as a single statement: Every element of X is paired with exactly one element of Y.Functions which satisfy property (3) are said to be "onto Y " and are called surjections (or surjective functions).Functions which satisfy property (4) are said to be "one-to-one functions" and are called injections (or injectivefunctions).[1] With this terminology, a bijection is a function which is both a surjection and an injection, or usingother words, a bijection is a function which is both one-to-one and onto.

1.2 Examples

1.2.1 Batting line-up of a baseball teamConsider the batting line-up of a baseball team (or any list of all the players of any sports team). The set X will be thenine players on the team and the set Y will be the nine positions in the batting order (1st, 2nd, 3rd, etc.) The pairingis given by which player is in what position in this order. Property (1) is satised since each player is somewhere inthe list. Property (2) is satised since no player bats in two (or more) positions in the order. Property (3) says thatfor each position in the order, there is some player batting in that position and property (4) states that two or moreplayers are never batting in the same position in the list.

1.2.2 Seats and students of a classroomIn a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks themall to be seated. After a quick look around the room, the instructor declares that there is a bijection between the setof students and the set of seats, where each student is paired with the seat they are sitting in. What the instructorobserved in order to reach this conclusion was that:

1. Every student was in a seat (there was no one standing),

2. No student was in more than one seat,

3. Every seat had someone sitting there (there were no empty seats), and

4. No seat had more than one student in it.

1.3. MORE MATHEMATICAL EXAMPLES AND SOME NON-EXAMPLES 3

The instructor was able to conclude that there were just as many seats as there were students, without having to counteither set.

1.3 More mathematical examples and some non-examples For any set X, the identity function 1X: X X, 1X(x) = x, is bijective. The function f: R R, f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y 1)/2 such that f(x)= y. In more generality, any linear function over the reals, f: R R, f(x) = ax + b (where a is non-zero) is abijection. Each real number y is obtained from (paired with) the real number x = (y - b)/a.

The function f: R (-/2, /2), given by f(x) = arctan(x) is bijective since each real number x is pairedwith exactly one angle y in the interval (-/2, /2) so that tan(y) = x (that is, y = arctan(x)). If the codomain(-/2, /2) was made larger to include an integer multiple of /2 then this function would no longer be onto(surjective) since there is no real number which could be paired with the multiple of /2 by this arctan function.

The exponential function, g: R R, g(x) = ex, is not bijective: for instance, there is no x in R such that g(x) =1, showing that g is not onto (surjective). However if the codomain is restricted to the positive real numbersR+ (0;+1) , then g becomes bijective; its inverse (see below) is the natural logarithm function ln.

The function h: R R+, h(x) = x2 is not bijective: for instance, h(1) = h(1) = 1, showing that h is not one-to-one (injective). However, if the domain is restricted to R+0 [0;+1) , then h becomes bijective; its inverseis the positive square root function.

1.4 InversesA bijection f with domainX (functionally indicated by f: XY) also denes a relation starting in Y and going toX(by turning the arrows around). The process of turning the arrows around for an arbitrary function does not usuallyyield a function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain Y.Moreover, properties (1) and (2) then say that this inverse function is a surjection and an injection, that is, the inversefunction exists and is also a bijection. Functions that have inverse functions are said to be invertible. A function isinvertible if and only if it is a bijection.Stated in concise mathematical notation, a function f: X Y is bijective if and only if it satises the condition

for every y in Y there is a unique x in X with y = f(x).

Continuing with the baseball batting line-up example, the function that is being dened takes as input the name ofone of the players and outputs the position of that player in the batting order. Since this function is a bijection, it hasan inverse function which takes as input a position in the batting order and outputs the player who will be batting inthat position.

1.5 CompositionThe composition g f of two bijections f: X Y and g: Y Z is a bijection. The inverse of g f is (g f)1 =(f1) (g1) .Conversely, if the composition g f of two functions is bijective, we can only say that f is injective and g is surjective.

1.6 Bijections and cardinalityIf X and Y are nite sets, then there exists a bijection between the two sets X and Y if and only if X and Y havethe same number of elements. Indeed, in axiomatic set theory, this is taken as the denition of same number ofelements (equinumerosity), and generalising this denition to innite sets leads to the concept of cardinal number,a way to distinguish the various sizes of innite sets.


X1

2

3

YD

B

C

A

ZP

Q

R

A bijection composed of an injection (left) and a surjection (right).

1.7 Properties A function f: R R is bijective if and only if its graph meets every horizontal and vertical line exactly once.

If X is a set, then the bijective functions from X to itself, together with the operation of functional composition(), form a group, the symmetric group of X, which is denoted variously by S(X), SX, or X! (X factorial).

Bijections preserve cardinalities of sets: for a subset A of the domain with cardinality |A| and subset B of thecodomain with cardinality |B|, one has the following equalities:

|f(A)| = |A| and |f1(B)| = |B|.

If X and Y are nite sets with the same cardinality, and f: X Y, then the following are equivalent:

1. f is a bijection.2. f is a surjection.3. f is an injection.

For a nite set S, there is a bijection between the set of possible total orderings of the elements and the set ofbijections from S to S. That is to say, the number of permutations of elements of S is the same as the numberof total orderings of that setnamely, n!.

1.8 Bijections and category theoryBijections are precisely the isomorphisms in the category Set of sets and set functions. However, the bijections are notalways the isomorphisms for more complex categories. For example, in the category Grp of groups, the morphismsmust be homomorphisms since they must preserve the group structure, so the isomorphisms are group isomorphismswhich are bijective homomorphisms.

1.9. GENERALIZATION TO PARTIAL FUNCTIONS 5

1.9 Generalization to partial functionsThe notion of one-one correspondence generalizes to partial functions, where they are called partial bijections,although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partialfunction is already undened for a portion of its domain; thus there is no compelling reason to constrain its inverseto be a total function, i.e. dened everywhere on its domain. The set of all partial bijections on a given base set iscalled the symmetric inverse semigroup.[2]

Another way of dening the same notion is to say that a partial bijection from A to B is any relation R (which turnsout to be a partial function) with the property that R is the graph of a bijection f:AB, where A is a subset of Aand likewise BB.[3]

When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation.[4] Anexample is theMbius transformation simply dened on the complex plane, rather than its completion to the extendedcomplex plane.[5]

1.10 Contrast withThis list is incomplete; you can help by expanding it.

Multivalued function

1.11 See also Injective function Surjective function Bijection, injection and surjection Symmetric group Bijective numeration Bijective proof Cardinality Category theory AxGrothendieck theorem

1.12 Notes[1] There are names associated to properties (1) and (2) as well. A relation which satises property (1) is called a total relation

and a relation satisfying (2) is a single valued relation.

[2] Christopher Hollings (16 July 2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups.American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.

[3] Francis Borceux (1994). Handbook of Categorical Algebra: Volume 2, Categories and Structures. Cambridge UniversityPress. p. 289. ISBN 978-0-521-44179-7.

[4] Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 228. ISBN 978-0-8247-9662-4.

[5] John Meakin (2007). Groups and semigroups: connections and contrasts. In C.M. Campbell, M.R. Quick, E.F. Robert-son, G.C. Smith. Groups St Andrews 2005 Volume 2. Cambridge University Press. p. 367. ISBN 978-0-521-69470-4.preprint citing Lawson,M.V. (1998). TheMbius InverseMonoid. Journal of Algebra 200 (2): 428. doi:10.1006/jabr.1997.7242.


1.13 ReferencesThis topic is a basic concept in set theory and can be found in any text which includes an introduction to set theory.Almost all texts that deal with an introduction to writing proofs will include a section on set theory, so the topic maybe found in any of these:

Wolf (1998). Proof, Logic and Conjecture: A Mathematicians Toolbox. Freeman. Sundstrom (2003). Mathematical Reasoning: Writing and Proof. Prentice-Hall. Smith; Eggen; St.Andre (2006). A Transition to Advanced Mathematics (6th Ed.). Thomson (Brooks/Cole). Schumacher (1996). Chapter Zero: Fundamental Notions of Abstract Mathematics. Addison-Wesley. O'Leary (2003). The Structure of Proof: With Logic and Set Theory. Prentice-Hall. Morash. Bridge to Abstract Mathematics. Random House. Maddox (2002). Mathematical Thinking and Writing. Harcourt/ Academic Press. Lay (2001). Analysis with an introduction to proof. Prentice Hall. Gilbert; Vanstone (2005). An Introduction to Mathematical Thinking. Pearson Prentice-Hall. Fletcher; Patty. Foundations of Higher Mathematics. PWS-Kent. Iglewicz; Stoyle. An Introduction to Mathematical Reasoning. MacMillan. Devlin, Keith (2004). Sets, Functions, and Logic: An Introduction to Abstract Mathematics. Chapman & Hall/CRC Press.

D'Angelo; West (2000). Mathematical Thinking: Problem Solving and Proofs. Prentice Hall. Cupillari. The Nuts and Bolts of Proofs. Wadsworth. Bond. Introduction to Abstract Mathematics. Brooks/Cole. Barnier; Feldman (2000). Introduction to Advanced Mathematics. Prentice Hall. Ash. A Primer of Abstract Mathematics. MAA.

1.14 External links Hazewinkel, Michiel, ed. (2001), Bijection, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Weisstein, Eric W., Bijection, MathWorld. Earliest Uses of Some of theWords ofMathematics: entry on Injection, Surjection and Bijection has the historyof Injection and related terms.

Chapter 2

Cardinal number

This article is about the mathematical concept. For number words indicating quantity (three apples, four birds,etc.), see Cardinal number (linguistics).In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to

measure the cardinality (size) of sets. The cardinality of a nite set is a natural number: the number of elements inthe set. The transnite cardinal numbers describe the sizes of innite sets.Cardinality is dened in terms of bijective functions. Two sets have the same cardinality if, and only if, there is aone-to-one correspondence (bijection) between the elements of the two sets. In the case of nite sets, this agrees withthe intuitive notion of size. In the case of innite sets, the behavior is more complex. A fundamental theorem due toGeorg Cantor shows that it is possible for innite sets to have dierent cardinalities, and in particular the cardinalityof the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a propersubset of an innite set to have the same cardinality as the original set, something that cannot happen with propersubsets of nite sets.There is a transnite sequence of cardinal numbers:

0; 1; 2; 3; : : : ; n; : : : ;@0;@1;@2; : : : ;@; : : : :This sequence starts with the natural numbers including zero (nite cardinals), which are followed by the alephnumbers (innite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal numbers. Under theassumption of the axiom of choice, this transnite sequence includes every cardinal number. If one rejects thataxiom, the situation is more complicated, with additional innite cardinals that are not alephs.Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics includingcombinatorics, abstract algebra, and mathematical analysis. In category theory, the cardinal numbers form a skeletonof the category of sets.

2.1 HistoryThe notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 18741884. Cardinality can be used to compare an aspect of nite sets; e.g. the sets {1,2,3} and {4,5,6} are not equal,but have the same cardinality, namely three (this is established by the existence of a bijection, i.e. a one-to-onecorrespondence, between the two sets; e.g. {1->4, 2->5, 3->6}).Cantor applied his concept of bijection to innite sets;[1] e.g. the set of natural numbers N = {0, 1, 2, 3, ...}. Thus,all sets having a bijection with N he called denumerable (countably innite) sets and they all have the same cardinalnumber. This cardinal number is called@0 , aleph-null. He called the cardinal numbers of these innite sets, transnitecardinal numbers.Cantor proved that any unbounded subset of N has the same cardinality as N, even though this might appear to runcontrary to intuition. He also proved that the set of all ordered pairs of natural numbers is denumerable (whichimplies that the set of all rational numbers is denumerable), and later proved that the set of all algebraic numbers isalso denumerable. Each algebraic number zmay be encoded as a nite sequence of integers which are the coecients

7

8 CHAPTER 2. CARDINAL NUMBER

X 1

2

3

4

YD

B

C

A

A bijective function, f: X Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to thecardinal number 4.

in the polynomial equation of which it is the solution, i.e. the ordered n-tuple (a0, a1, ..., an), ai Z together with apair of rationals (b0, b1) such that z is the unique root of the polynomial with coecients (a0, a1, ..., an) that lies inthe interval (b0, b1).In his 1874 paper, Cantor proved that there exist higher-order cardinal numbers by showing that the set of real numbershas cardinality greater than that of N. His original presentation used a complex argument with nested intervals, but inan 1891 paper he proved the same result using his ingenious but simple diagonal argument. The new cardinal numberof the set of real numbers is called the cardinality of the continuum and Cantor used the symbol c for it.Cantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallesttransnite cardinal number ( @0 , aleph-null) and that for every cardinal number, there is a next-larger cardinal

(@1;@2;@3; ):

His continuum hypothesis is the proposition that c is the same as @1 . This hypothesis has been found to be inde-pendent of the standard axioms of mathematical set theory; it can neither be proved nor disproved from the standardassumptions.

2.2. MOTIVATION 9

Aleph null, the smallest innite cardinal

2.2 Motivation

In informal use, a cardinal number is what is normally referred to as a counting number, provided that 0 is included:0, 1, 2, .... They may be identied with the natural numbers beginning with 0. The counting numbers are exactlywhat can be dened formally as the nite cardinal numbers. Innite cardinals only occur in higher-level mathematicsand logic.More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe theposition of an element in a sequence. For nite sets and sequences it is easy to see that these two notions coincide,since for every number describing a position in a sequence we can construct a set which has exactly the right size,e.g. 3 describes the position of 'c' in the sequence , and we can construct the set {a,b,c} which has3 elements. However when dealing with innite sets it is essential to distinguish between the two the two notionsare in fact dierent for innite sets. Considering the position aspect leads to ordinal numbers, while the size aspect isgeneralized by the cardinal numbers described here.The intuition behind the formal denition of cardinal is the construction of a notion of the relative size or bigness ofa set without reference to the kind of members which it has. For nite sets this is easy; one simply counts the numberof elements a set has. In order to compare the sizes of larger sets, it is necessary to appeal to more subtle notions.


A set Y is at least as big as a set X if there is an injective mapping from the elements of X to the elements of Y.An injective mapping identies each element of the set X with a unique element of the set Y. This is most easilyunderstood by an example; suppose we have the sets X = {1,2,3} and Y = {a,b,c,d}, then using this notion of size wewould observe that there is a mapping:

1 a2 b3 c

which is injective, and hence conclude that Y has cardinality greater than or equal to X. Note the element d has noelement mapping to it, but this is permitted as we only require an injective mapping, and not necessarily an injectiveand onto mapping. The advantage of this notion is that it can be extended to innite sets.We can then extend this to an equality-style relation. Two sets X and Y are said to have the same cardinality if thereexists a bijection between X and Y. By the SchroederBernstein theorem, this is equivalent to there being both aninjective mapping from X to Y and an injective mapping from Y to X. We then write |X| = |Y |. The cardinal numberofX itself is often dened as the least ordinal awith |a| = |X|. This is called the von Neumann cardinal assignment; forthis denition to make sense, it must be proved that every set has the same cardinality as some ordinal; this statement isthe well-ordering principle. It is however possible to discuss the relative cardinality of sets without explicitly assigningnames to objects.The classic example used is that of the innite hotel paradox, also called Hilberts paradox of the Grand Hotel.Suppose you are an innkeeper at a hotel with an innite number of rooms. The hotel is full, and then a new guestarrives. It is possible to t the extra guest in by asking the guest who was in room 1 to move to room 2, the guest inroom 2 to move to room 3, and so on, leaving room 1 vacant. We can explicitly write a segment of this mapping:

1 22 33 4...n n + 1...

In this way we can see that the set {1,2,3,...} has the same cardinality as the set {2,3,4,...} since a bijection betweenthe rst and the second has been shown. This motivates the denition of an innite set being any set which has aproper subset of the same cardinality; in this case {2,3,4,...} is a proper subset of {1,2,3,...}.When considering these large objects, we might also want to see if the notion of counting order coincides with thatof cardinal dened above for these innite sets. It happens that it doesn't; by considering the above example we cansee that if some object one greater than innity exists, then it must have the same cardinality as the innite setwe started out with. It is possible to use a dierent formal notion for number, called ordinals, based on the ideasof counting and considering each number in turn, and we discover that the notions of cardinality and ordinality aredivergent once we move out of the nite numbers.It can be proved that the cardinality of the real numbers is greater than that of the natural numbers just described.This can be visualized using Cantors diagonal argument; classic questions of cardinality (for instance the continuumhypothesis) are concerned with discovering whether there is some cardinal between some pair of other innite car-dinals. In more recent times mathematicians have been describing the properties of larger and larger cardinals.Since cardinality is such a common concept in mathematics, a variety of names are in use. Sameness of cardinality issometimes referred to as equipotence, equipollence, or equinumerosity. It is thus said that two sets with the samecardinality are, respectively, equipotent, equipollent, or equinumerous.

2.3 Formal denitionFormally, assuming the axiom of choice, the cardinality of a set X is the least ordinal such that there is a bijectionbetween X and . This denition is known as the von Neumann cardinal assignment. If the axiom of choice is not

2.4. CARDINAL ARITHMETIC 11

assumed we need to do something dierent. The oldest denition of the cardinality of a set X (implicit in Cantor andexplicit in Frege and Principia Mathematica) is as the class [X] of all sets that are equinumerous with X. This doesnot work in ZFC or other related systems of axiomatic set theory because if X is non-empty, this collection is toolarge to be a set. In fact, for X there is an injection from the universe into [X] by mapping a set m to {m} Xand so by the axiom of limitation of size, [X] is a proper class. The denition does work however in type theory andin New Foundations and related systems. However, if we restrict from this class to those equinumerous with X thathave the least rank, then it will work (this is a trick due to Dana Scott:[2] it works because the collection of objectswith any given rank is a set).Formally, the order among cardinal numbers is dened as follows: |X| |Y | means that there exists an injectivefunction from X to Y. The CantorBernsteinSchroeder theorem states that if |X| |Y | and |Y | |X| then |X| = |Y |.The axiom of choice is equivalent to the statement that given two sets X and Y, either |X| |Y | or |Y | |X|.[3][4]

A set X is Dedekind-innite if there exists a proper subset Y of X with |X| = |Y |, and Dedekind-nite if such a subsetdoesn't exist. The nite cardinals are just the natural numbers, i.e., a set X is nite if and only if |X| = |n| = n forsome natural number n. Any other set is innite. Assuming the axiom of choice, it can be proved that the Dedekindnotions correspond to the standard ones. It can also be proved that the cardinal @0 (aleph null or aleph-0, where alephis the rst letter in the Hebrew alphabet, represented @ ) of the set of natural numbers is the smallest innite cardinal,i.e. that any innite set has a subset of cardinality @0: The next larger cardinal is denoted by @1 and so on. For everyordinal there is a cardinal number @; and this list exhausts all innite cardinal numbers.

2.4 Cardinal arithmeticWe can dene arithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers.It can be shown that for nite cardinals these operations coincide with the usual operations for natural numbers.Furthermore, these operations share many properties with ordinary arithmetic.

2.4.1 Successor cardinalFor more details on this topic, see Successor cardinal.

If the axiom of choice holds, every cardinal has a successor + > , and there are no cardinals between and itssuccessor. (Without the axiom of choice, using Hartogs theorem, it can be shown that, for any cardinal number ,there is a minimal cardinal +, so that + : ) For nite cardinals, the successor is simply + 1. For innitecardinals, the successor cardinal diers from the successor ordinal.

2.4.2 Cardinal additionIf X and Y are disjoint, addition is given by the union of X and Y. If the two sets are not already disjoint, then theycan be replaced by disjoint sets of the same cardinality, e.g., replace X by X{0} and Y by Y{1}.

jXj+ jY j = jX [ Y j:Zero is an additive identity + 0 = 0 + = .Addition is associative ( + ) + = + ( + ).Addition is commutative + = + .Addition is non-decreasing in both arguments:

( ) ! ((+ + ) and ( + + )):Assuming the axiom of choice, addition of innite cardinal numbers is easy. If either or is innite, then

+ = maxf; g :


Subtraction

Assuming the axiom of choice and, given an innite cardinal and a cardinal , there exists a cardinal such that + = if and only if . It will be unique (and equal to ) if and only if < .

2.4.3 Cardinal multiplicationThe product of cardinals comes from the cartesian product.

jXj jY j = jX Y j0 = 0 = 0. = 0 ( = 0 or = 0).One is a multiplicative identity 1 = 1 = .Multiplication is associative () = ().Multiplication is commutative = .Multiplication is non-decreasing in both arguments: ( and ).Multiplication distributes over addition: ( + ) = + and ( + ) = + .Assuming the axiom of choice, multiplication of innite cardinal numbers is also easy. If either or is innite andboth are non-zero, then

= maxf; g:

Division

Assuming the axiom of choice and, given an innite cardinal and a non-zero cardinal , there exists a cardinal such that = if and only if . It will be unique (and equal to ) if and only if < .

2.4.4 Cardinal exponentiationExponentiation is given by

jXjjY j = XY where XY is the set of all functions from Y to X.

0 = 1 (in particular 00 = 1), see empty function.If 1 , then 0 = 0.1 = 1.1 = . + = . = ().() = .

Exponentiation is non-decreasing in both arguments:

(1 and ) ( ) and( ) ( ).

2.5. THE CONTINUUM HYPOTHESIS 13

Note that 2|X| is the cardinality of the power set of the set X and Cantors diagonal argument shows that 2|X| > |X| forany set X. This proves that no largest cardinal exists (because for any cardinal , we can always nd a larger cardinal2). In fact, the class of cardinals is a proper class. (This proof fails in some set theories, notably New Foundations.)All the remaining propositions in this section assume the axiom of choice:

If and are both nite and greater than 1, and is innite, then = .If is innite and is nite and non-zero, then = .

If 2 and 1 and at least one of them is innite, then:

Max (, 2) Max (2, 2).

Using Knigs theorem, one can prove < cf() and < cf(2) for any innite cardinal , where cf() is the conalityof .

Roots

Assuming the axiom of choice and, given an innite cardinal and a nite cardinal greater than 0, the cardinal satisfying = will be .

Logarithms

Assuming the axiom of choice and, given an innite cardinal and a nite cardinal greater than 1, there may ormay not be a cardinal satisfying = . However, if such a cardinal exists, it is innite and less than , and anynite cardinality greater than 1 will also satisfy = .The logarithm of an innite cardinal number is dened as the least cardinal number such that 2. Logarithmsof innite cardinals are useful in some elds of mathematics, for example in the study of cardinal invariants oftopological spaces, though they lack some of the properties that logarithms of positive real numbers possess.[5][6][7]

2.5 The continuum hypothesisThe continuum hypothesis (CH) states that there are no cardinals strictly between @0 and 2@0 : The latter cardinalnumber is also often denoted by c ; it is the cardinality of the continuum (the set of real numbers). In this case2@0 = @1: The generalized continuum hypothesis (GCH) states that for every innite set X, there are no cardinalsstrictly between | X | and 2| X |. The continuum hypothesis is independent of the usual axioms of set theory, theZermelo-Fraenkel axioms together with the axiom of choice (ZFC).

2.6 See also

2.7 Notes

2.8 ReferencesNotes

[1] Dauben 1990, pg. 54

[2] Deiser, Oliver (May 2010). On the Development of the Notion of a Cardinal Number. History and Philosophy of Logic31 (2): 123143. doi:10.1080/01445340903545904.

[3] Enderton, Herbert. Elements of Set Theory, Academic Press Inc., 1977. ISBN 0-12-238440-7


[4] Friedrich M. Hartogs (1915), Felix Klein, Walther von Dyck, David Hilbert, Otto Blumenthal, ed., "ber das Problem derWohlordnung, Math. Ann (Leipzig: B. G. Teubner), Bd. 76 (4): 438443, ISSN 0025-5831

[5] Robert A. McCoy and Ibula Ntantu, Topological Properties of Spaces of Continuous Functions, Lecture Notes in Mathe-matics 1315, Springer-Verlag.

[6] Eduard ech, Topological Spaces, revised by Zdenek Frolk and Miroslav Katetov, John Wiley & Sons, 1966.

[7] D.A. Vladimirov, Boolean Algebras in Analysis, Mathematics and Its Applications, Kluwer Academic Publishers.

Bibliography

Dauben, Joseph Warren (1990), Georg Cantor: His Mathematics and Philosophy of the Innite, Princeton:Princeton University Press, ISBN 0691-02447-2

Hahn, Hans, Innity, Part IX, Chapter 2, Volume 3 of The World of Mathematics. New York: Simon andSchuster, 1956.

Halmos, Paul, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).

2.9 External links Hazewinkel, Michiel, ed. (2001), Cardinal number, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Weisstein, Eric W., Cardinal Number, MathWorld. Cardinality at ProvenMath proofs of the basic theorems on cardinality.

Chapter 3

Cardinality

For other uses, see Cardinality (disambiguation).

In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the setA = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3. There are two approaches to cardinality one which compares sets directly using bijections and injections, and another which uses cardinal numbers.[1] Thecardinality of a set is also called its size, when no confusion with other notions of size[2] is possible.The cardinality of a set A is usually denoted | A |, with a vertical bar on each side; this is the same notation as absolutevalue and the meaning depends on context. Alternatively, the cardinality of a set A may be denoted by n(A), A,card(A), or # A.

3.1 Comparing setsWhile the cardinality of a nite set is just the number of its elements, extending the notion to innite sets usuallystarts with dening the notion of comparison of arbitrary (in particular innite) sets.

3.1.1 Denition 1: | A | = | B |Two sets A and B have the same cardinality if there exists a bijection, that is, an injective and surjectivefunction, from A to B. Such sets are said to be equipotent, equipollent, or equinumerous.

For example, the set E = {0, 2, 4, 6, ...} of non-negative even numbers has the same cardinality as theset N = {0, 1, 2, 3, ...} of natural numbers, since the function f(n) = 2n is a bijection from N to E.

3.1.2 Denition 2: | A | | B |A has cardinality less than or equal to the cardinality of B if there exists an injective function from Ainto B.

3.1.3 Denition 3: | A | < | B |A has cardinality strictly less than the cardinality of B if there is an injective function, but no bijectivefunction, from A to B.

For example, the setN of all natural numbers has cardinality strictly less than the cardinality of the setRof all real numbers , because the inclusion map i : NR is injective, but it can be shown that there doesnot exist a bijective function fromN toR (see Cantors diagonal argument or Cantors rst uncountabilityproof).

15

16 CHAPTER 3. CARDINALITY

EN

012

n

024

2n. .

. .

Bijective function from N to E. Although E is a proper subset of N, both sets have the same cardinality.

If | A | | B | and | B | | A | then | A | = | B | (CantorBernsteinSchroeder theorem). The axiom of choice isequivalent to the statement that | A | | B | or | B | | A | for every A,B.[3][4]

3.2 Cardinal numbersMain article: Cardinal number

Above, cardinality was dened functionally. That is, the cardinality of a set was not dened as a specic objectitself. However, such an object can be dened as follows.The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the classof all sets. The equivalence class of a set A under this relation then consists of all those sets which have the samecardinality as A. There are two ways to dene the cardinality of a set":

1. The cardinality of a set A is dened as its equivalence class under equinumerosity.2. A representative set is designated for each equivalence class. The most common choice is the initial ordinal in

that class. This is usually taken as the denition of cardinal number in axiomatic set theory.

Assuming AC, the cardinalities of the innite sets are denoted

@0 < @1 < @2 < : : : :For each ordinal , @+1 is the least cardinal number greater than @ .

3.3. FINITE, COUNTABLE AND UNCOUNTABLE SETS 17

The cardinality of the natural numbers is denoted aleph-null (@0 ), while the cardinality of the real numbers is denotedby " c " (a lowercase fraktur script c), and is also referred to as the cardinality of the continuum. Cantor showed,using the diagonal argument, that c > @0 . We can show that c = 2@0 , this also being the cardinality of the set ofall subsets of the natural numbers. The continuum hypothesis says that @1 = 2@0 , i.e. 2@0 is the smallest cardinalnumber bigger than @0 , i.e. there is no set whose cardinality is strictly between that of the integers and that of thereal numbers. The continuum hypothesis is independent of ZFC, a standard axiomatization of set theory; that is, it isimpossible to prove the continuum hypothesis or its negation from ZFC (provided ZFC is consistent). See below formore details on the cardinality of the continuum.[5][6][7]

3.3 Finite, countable and uncountable setsIf the axiom of choice holds, the law of trichotomy holds for cardinality. Thus we can make the following denitions:

Any set X with cardinality less than that of the natural numbers, or | X | < | N |, is said to be a nite set. Any set X that has the same cardinality as the set of the natural numbers, or | X | = | N | = @0 , is said to be acountably innite set.

Any set X with cardinality greater than that of the natural numbers, or | X | > | N |, for example | R | = c > | N|, is said to be uncountable.

3.4 Innite setsOur intuition gained from nite sets breaks down when dealing with innite sets. In the late nineteenth century GeorgCantor, Gottlob Frege, Richard Dedekind and others rejected the view of Galileo (which derived from Euclid) thatthe whole cannot be the same size as the part. One example of this is Hilberts paradox of the Grand Hotel. Indeed,Dedekind dened an innite set as one that can be placed into a one-to-one correspondence with a strict subset (thatis, having the same size in Cantors sense); this notion of innity is called Dedekind innite. Cantor introduced thecardinal numbers, and showed that (according to his bijection-based denition of size) some innite sets are greaterthan others. The smallest innite cardinality is that of the natural numbers ( @0 ).

3.4.1 Cardinality of the continuumMain article: Cardinality of the continuum

One of Cantors most important results was that the cardinality of the continuum ( c ) is greater than that of the naturalnumbers ( @0 ); that is, there are more real numbers R than whole numbers N. Namely, Cantor showed that

c = 2@0 > @0(see Cantors diagonal argument or Cantors rst uncountability proof).

The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardi-nality of the natural numbers, that is,

c = @1 = i1(see Beth one).

However, this hypothesis can neither be proved nor disproved within the widely accepted ZFC axiomatic set theory,if ZFC is consistent.Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to thenumber of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed,in any nite-dimensional space. These results are highly counterintuitive, because they imply that there exist proper

18 CHAPTER 3. CARDINALITY

subsets and proper supersets of an innite set S that have the same size as S, although S contains elements that do notbelong to its subsets, and the supersets of S contain elements that are not included in it.The rst of these results is apparent by considering, for instance, the tangent function, which provides a one-to-onecorrespondence between the interval (, ) and R (see also Hilberts paradox of the Grand Hotel).The second result was rst demonstrated by Cantor in 1878, but it became more apparent in 1890, when GiuseppePeano introduced the space-lling curves, curved lines that twist and turn enough to ll the whole of any square, orcube, or hypercube, or nite-dimensional space. These curves are not a direct proof that a line has the same numberof points as a nite-dimensional space, but they can be used to obtain such a proof.Cantor also showed that sets with cardinality strictly greater than c exist (see his generalized diagonal argument andtheorem). They include, for instance:

the set of all subsets of R, i.e., the power set of R, written P(R) or 2R the set RR of all functions from R to R

Both have cardinality

2c = i2 > c

(see Beth two).

The cardinal equalities c2 = c; c@0 = c; and cc = 2c can be demonstrated using cardinal arithmetic:

c2 =2@02

= 22@0 = 2@0 = c;

c@0 =2@0@0

= 2@0@0 = 2@0 = c;

cc =2@0c

= 2c@0 = 2c:

3.5 Examples and properties If X = {a, b, c} and Y = {apples, oranges, peaches}, then | X | = | Y | because { (a, apples), (b, oranges), (c,peaches)} is a bijection between the sets X and Y. The cardinality of each of X and Y is 3.

If | X | < | Y |, then there exists Z such that | X | = | Z | and Z Y.

If | X | | Y | and | Y | | X |, then | X | = | Y |. This holds even for innite cardinals, and is known asCantorBernsteinSchroeder theorem.

Sets with cardinality of the continuum

3.6 Union and intersectionIf A and B are disjoint sets, then

jA [Bj = jAj+ jBj :

From this, one can show that in general the cardinalities of unions and intersections are related by[8]

jC [Dj+ jC \Dj = jCj+ jDj :

3.7. SEE ALSO 19

3.7 See also Aleph number Beth number Countable set Ordinality

3.8 References[1] Weisstein, Eric W., Cardinal Number, MathWorld.

[2] Such as length and area in geometry. A line of nite length is a set of points that has innite cardinality.

[3] Friedrich M. Hartogs (1915), Felix Klein, Walther von Dyck, David Hilbert, Otto Blumenthal, ed., "ber das Problemder Wohlordnung, Mathematische Annalen (Leipzig: B. G. Teubner) 76 (4): 438443, doi:10.1007/bf01458215, ISSN0025-5831

[4] FelixHausdor (2002), Egbert Brieskorn, Srishti D. Chatterji et al., eds.,Grundzge derMengenlehre (1. ed.), Berlin/Heidelberg:Springer, p. 587, ISBN 3-540-42224-2 - Original edition (1914)

[5] Cohen, Paul J. (December 15, 1963). The Independence of the Continuum Hypothesis. Proceedings of the NationalAcademy of Sciences of the United States of America 50 (6): 11431148. doi:10.1073/pnas.50.6.1143. JSTOR 71858.PMC 221287. PMID 16578557.

[6] Cohen, Paul J. (January 15, 1964). The Independence of the Continuum Hypothesis, II. Proceedings of the NationalAcademy of Sciences of the United States of America 51 (1): 105110. doi:10.1073/pnas.51.1.105. JSTOR 72252. PMC300611. PMID 16591132.

[7] Penrose, R (2005), The Road to Reality: A Complete guide to the Laws of the Universe, Vintage Books, ISBN 0-09-944068-7

[8] Applied Abstract Algebra, K.H. Kim, F.W. Roush, Ellis Horwood Series, 1983, ISBN 0-85312-612-7 (student edition),ISBN 0-85312-563-5 (library edition)

Chapter 4

Coniteness

Not to be confused with conality.

In mathematics, a conite subset of a set X is a subset A whose complement in X is a nite set. In other words, Acontains all but nitely many elements of X. If the complement is not nite, but it is countable, then one says the setis cocountable.These arise naturally when generalizing structures on nite sets to innite sets, particularly on innite products, as inthe product topology or direct sum.

4.1 Boolean algebrasThe set of all subsets ofX that are either nite or conite forms a Boolean algebra, i.e., it is closed under the operationsof union, intersection, and complementation. This Boolean algebra is the nite-conite algebra on X. A Booleanalgebra A has a unique non-principal ultralter (i.e. a maximal lter not generated by a single element of the algebra)if and only if there is an innite set X such that A is isomorphic to the nite-conite algebra on X. In this case, thenon-principal ultralter is the set of all conite sets.

4.2 Conite topologyThe conite topology (sometimes called the nite complement topology) is a topology which can be dened onevery set X. It has precisely the empty set and all conite subsets of X as open sets. As a consequence, in the conitetopology, the only closed subsets are nite sets, or the whole of X. Symbolically, one writes the topology as

T = fA X j A = ? or X nA is niteg

This topology occurs naturally in the context of the Zariski topology. Since polynomials over a eld K are zero onnite sets, or the whole of K, the Zariski topology on K (considered as ane line) is the conite topology. The sameis true for any irreducible algebraic curve; it is not true, for example, for XY = 0 in the plane.

4.2.1 Properties Subspaces: Every subspace topology of the conite topology is also a conite topology. Compactness: Since every open set contains all but nitely many points of X, the space X is compact andsequentially compact.

Separation: The conite topology is the coarsest topology satisfying the T1 axiom; i.e. it is the smallest topologyfor which every singleton set is closed. In fact, an arbitrary topology on X satises the T1 axiom if and only if

20

4.3. OTHER EXAMPLES 21

it contains the conite topology. If X is nite then the conite topology is simply the discrete topology. If X isnot nite, then this topology is not T2, regular or normal, since no two nonempty open sets are disjoint (i.e. itis hyperconnected).

4.2.2 Double-pointed conite topologyThe double-pointed conite topology is the conite topology with every point doubled; that is, it is the topologicalproduct of the conite topology with the indiscrete topology. It is not T0 or T1, since the points of the doublet aretopologically indistinguishable. It is, however, R0 since the topologically distinguishable points are separable.An example of a countable double-pointed conite topology is the set of even and odd integers, with a topology thatgroups them together. Let X be the set of integers, and let OA be a subset of the integers whose complement is the setA. Dene a subbase of open sets Gx for any integer x to be Gx = O{x, x} if x is an even number, and Gx = O{x,x} if x is odd. Then the basis sets of X are generated by nite intersections, that is, for nite A, the open sets of thetopology are

UA :=\x2A

Gx

The resulting space is not T0 (and hence not T1), because the points x and x + 1 (for x even) are topologicallyindistinguishable. The space is, however, a compact space, since it is covered by a nite union of the UA.

4.3 Other examples

4.3.1 Product topologyThe product topology on a product of topological spacesQXi has basisQUi where Ui Xi is open, and conitelymany Ui = Xi .The analog (without requiring that conitely many are the whole space) is the box topology.

4.3.2 Direct sumThe elements of the direct sum of modulesLMi are sequences i 2Mi where conitely many i = 0 .The analog (without requiring that conitely many are zero) is the direct product.

4.4 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 18)

Chapter 5

Derived set (mathematics)

In mathematics, more specically in point-set topology, the derived set of a subset S of a topological space is the setof all limit points of S. It is usually denoted by S0 .The concept was rst introduced by Georg Cantor in 1872 and he developed set theory in large part to study derivedsets on the real line.

5.1 PropertiesA subset S of a topological space is closed precisely when S0 S , when S contains all its limit points. Two subsetsS and T are separated precisely when they are disjoint and each is disjoint from the others derived set (though thederived sets don't need to be disjoint from each other).The set S is dened to be a perfect set if S = S0 . Equivalently, a perfect set is a closed set with no isolated points.Perfect sets are particularly important in applications of the Baire category theorem.Two topological spaces are homeomorphic if and only if there is a bijection from one to the other such that the derivedset of the image of any subset is the image of the derived set of that subset.The CantorBendixson theorem states that any Polish space can be written as the union of a countable set and aperfect set. Because any G subset of a Polish space is again a Polish space, the theorem also shows that any Gsubset of a Polish space is the union of a countable set and a set that is perfect with respect to the induced topology.

5.2 Topology in terms of derived setsBecause homeomorphisms can be described entirely in terms of derived sets, derived sets have been used as theprimitive notion in topology. A set of points X can be equipped with an operator * mapping subsets of X to subsetsof X, such that for any set S and any point a:

1. ; = ;2. S S

3. a 2 S =) a 2 (S n fag)

4. (S [ T ) S [ T

5. S T =) S T

Note that given 5, 3 is equivalent to 3' below, and that 4 and 5 together are equivalent to 4' below, so we have thefollowing equivalent axioms:

1. ; = ;

22

5.3. CANTORBENDIXSON RANK 23

2. S S

3'. S = (S n fag)

4'. (S [ T ) = S [ T

Calling a set S closed if S S will dene a topology on the space in which * is the derived set operator, that is,S = S0 . If we also require that the derived set of a set consisting of a single element be empty, the resulting spacewill be a T1 space. In fact, 2 and 3' can fail in a space that is not T1.

5.3 CantorBendixson rankFor ordinal numbers , the -th CantorBendixson derivative of a topological space is dened by transnite in-duction as follows:

X0 = X X+1 = (X)0

X =\

Chapter 6

Discrete space

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in whichthe points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. The discretetopology is the nest topology that can be given on a set, i.e., it denes all subsets as open sets. In particular, eachsingleton is an open set in the discrete topology.

6.1 DenitionsGiven a set X:

the discrete topology on X is dened by letting every subset of X be open (and hence also closed), and X is adiscrete topological space if it is equipped with its discrete topology;

the discrete uniformity on X is dened by letting every superset of the diagonal {(x,x) : x is in X} in X Xbe an entourage, and X is a discrete uniform space if it is equipped with its discrete uniformity.

the discrete metric on X is dened by

(x; y) =

1 if x 6= y;0 if x = y

for any x; y 2 X . In this case (X; ) is called a discrete metric space or a space of isolated points.

a set S is discrete in a metric space (X; d) , for S X , if for every x 2 S , there exists some > 0(depending on x ) such that d(x; y) > for all y 2 S n fxg ; such a set consists of isolated points. A set S isuniformly discrete in the metric space (X; d) , for S X , if there exists > 0 such that for any two distinctx; y 2 S , d(x; y) > .

A metric space (E; d) is said to be uniformly discrete if there exists a packing radius r > 0 such that, for anyx; y 2 E , one has either x = y or d(x; y) > r .[1] The topology underlying a metric space can be discrete, withoutthe metric being uniformly discrete: for example the usual metric on the set {1, 1/2, 1/4, 1/8, ...} of real numbers.

6.2 PropertiesThe underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on adiscrete uniform space is the discrete topology. Thus, the dierent notions of discrete space are compatible with oneanother. On the other hand, the underlying topology of a non-discrete uniform or metric space can be discrete; anexample is the metric space X := {1/n : n = 1,2,3,...} (with metric inherited from the real line and given by d(x,y) =|x y|). Obviously, this is not the discrete metric; also, this space is not complete and hence not discrete as a uniformspace. Nevertheless, it is discrete as a topological space. We say that X is topologically discrete but not uniformlydiscrete or metrically discrete.Additionally:

24

6.2. PROPERTIES 25

The topological dimension of a discrete space is equal to 0.

A topological space is discrete if and only if its singletons are open, which is the case if and only if it doesn'tcontain any accumulation points.

The singletons form a basis for the discrete topology.

A uniform space X is discrete if and only if the diagonal {(x,x) : x is in X} is an entourage.

Every discrete topological space satises each of the separation axioms; in particular, every discrete space isHausdor, that is, separated.

A discrete space is compact if and only if it is nite.

Every discrete uniform or metric space is complete.

Combining the above two facts, every discrete uniform or metric space is totally bounded if and only if it isnite.

Every discrete metric space is bounded.

Every discrete space is rst-countable; it is moreover second-countable if and only if it is countable.

Every discrete space with at least two points is totally disconnected.

Every non-empty discrete space is second category.

Any two discrete spaces with the same cardinality are homeomorphic.

Every discrete space is metrizable (by the discrete metric).

A nite space is metrizable only if it is discrete.

If X is a topological space and Y is a set carrying the discrete topology, then X is evenly covered by X Y (theprojection map is the desired covering)

The subspace topology on the integers as a subspace of the real line is the discrete topology.

A discrete space is separable if and only if it is countable.

Any function from a discrete topological space to another topological space is continuous, and any function from adiscrete uniform space to another uniform space is uniformly continuous. That is, the discrete space X is free on theset X in the category of topological spaces and continuous maps or in the category of uniform spaces and uniformlycontinuous maps. These facts are examples of a much broader phenomenon, in which discrete structures are usuallyfree on sets.With metric spaces, things are more complicated, because there are several categories of metric spaces, depending onwhat is chosen for the morphisms. Certainly the discrete metric space is free when the morphisms are all uniformlycontinuous maps or all continuous maps, but this says nothing interesting about the metric structure, only the uniformor topological structure. Categories more relevant to the metric structure can be found by limiting the morphismsto Lipschitz continuous maps or to short maps; however, these categories don't have free objects (on more thanone element). However, the discrete metric space is free in the category of bounded metric spaces and Lipschitzcontinuous maps, and it is free in the category of metric spaces bounded by 1 and short maps. That is, any functionfrom a discrete metric space to another boundedmetric space is Lipschitz continuous, and any function from a discretemetric space to another metric space bounded by 1 is short.Going the other direction, a function f from a topological space Y to a discrete space X is continuous if and only ifit is locally constant in the sense that every point in Y has a neighborhood on which f is constant.

26 CHAPTER 6. DISCRETE SPACE

6.3 UsesA discrete structure is often used as the default structure on a set that doesn't carry any other natural topology,uniformity, or metric; discrete structures can often be used as extreme examples to test particular suppositions.For example, any group can be considered as a topological group by giving it the discrete topology, implying thattheorems about topological groups apply to all groups. Indeed, analysts may refer to the ordinary, non-topologicalgroups studied by algebraists as "discrete groups" . In some cases, this can be usefully applied, for example incombination with Pontryagin duality. A 0-dimensional manifold (or dierentiable or analytical manifold) is nothingbut a discrete topological space. We can therefore view any discrete group as a 0-dimensional Lie group.A product of countably innite copies of the discrete space of natural numbers is homeomorphic to the space ofirrational numbers, with the homeomorphism given by the continued fraction expansion. A product of countablyinnite copies of the discrete space {0,1} is homeomorphic to the Cantor set; and in fact uniformly homeomorphicto the Cantor set if we use the product uniformity on the product. Such a homeomorphism is given by using ternarynotation of numbers. (See Cantor space.)In the foundations of mathematics, the study of compactness properties of products of {0,1} is central to the topo-logical approach to the ultralter principle, which is a weak form of choice.

6.4 Indiscrete spacesMain article: Trivial topology

In some ways, the opposite of the discrete topology is the trivial topology (also called the indiscrete topology), whichhas the fewest possible open sets (just the empty set and the space itself). Where the discrete topology is initialor free, the indiscrete topology is nal or cofree: every function from a topological space to an indiscrete space iscontinuous, etc.

6.5 Quotation Stanislaw Ulam characterized Los Angeles as a discrete space, in which there is an hours drive betweenpoints.[2]

6.6

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