Partially Ordered Set

Partially ordered setFrom Wikipedia, the free encyclopedia

Contents

1 a-paracompact space 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Binary relation 22.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Special types of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2.1 Difunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Relations over a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Operations on binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.2 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.3 Algebras, categories, and rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Sets versus classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 The number of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.7 Examples of common binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Closure (topology) 123.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1.1 Point of closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1.2 Limit point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1.3 Closure of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Closure operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Facts about closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.5 Categorical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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3.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Compact operator 164.1 Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Important properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3 Origins in integral equation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.4 Compact operator on Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.5 Completely continuous operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Compact space 205.1 Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.3.1 Open cover denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3.2 Equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3.3 Compactness of subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.4 Properties of compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.4.1 Functions and compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.4.2 Compact spaces and set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.4.3 Ordered compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.5.1 Algebraic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 Compactly embedded 296.1 Denition (topological spaces) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.2 Denition (normed spaces) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7 Cover (topology) 307.1 Cover in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.2 Renement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.4 Covering dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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7.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

8 Exhaustion by compact sets 338.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

9 Feebly compact space 34

10 Functional analysis 3510.1 Normed vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10.1.1 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.1.2 Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10.2 Major and foundational results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.2.1 Uniform boundedness principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.2.2 Spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.2.3 Hahn-Banach theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.2.4 Open mapping theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.2.5 Closed graph theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.2.6 Other topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

10.3 Foundations of mathematics considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.4 Points of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

11 H-closed space 4111.1 Examples and equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

12 Hasse diagram 4212.1 A good Hasse diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.2 Upward planarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

13 Hemicompact space 4613.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4613.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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13.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4613.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

14 Interior (topology) 4814.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

14.1.1 Interior point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.1.2 Interior of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

14.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.3 Interior operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.4 Exterior of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.5 Interior-disjoint shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

15 k-cell (mathematics) 5315.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5315.2 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5315.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

16 Lebesgue covering dimension 5516.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5516.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5516.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5516.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5616.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

16.5.1 Historical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5616.5.2 Modern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

16.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5616.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

17 Limit point compact 5717.1 Properties and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5717.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5717.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5817.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

18 Lindelf space 5918.1 Properties of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5918.2 Properties of strongly Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5918.3 Product of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5918.4 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6018.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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18.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6018.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

19 Locally compact space 6119.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6119.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

19.2.1 Compact Hausdor spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6219.2.2 Locally compact Hausdor spaces that are not compact . . . . . . . . . . . . . . . . . . . 6219.2.3 Hausdor spaces that are not locally compact . . . . . . . . . . . . . . . . . . . . . . . . 6219.2.4 Non-Hausdor examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

19.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6319.3.1 The point at innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6319.3.2 Locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

19.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6419.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

20 Locally nite 65

21 Locally nite collection 6621.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

21.1.1 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6621.1.2 Second countable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

21.2 Closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6721.3 Countably locally nite collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6721.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

22 Locally nite space 6822.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

23 Mesocompact space 6923.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6923.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

24 Metacompact space 7024.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7024.2 Covering dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7024.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7024.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

25 Order theory 7225.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7225.2 Basic denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

25.2.1 Partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7325.2.2 Visualizing a poset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

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25.2.3 Special elements within an order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7325.2.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7525.2.5 Constructing new orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

25.3 Functions between orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7525.4 Special types of orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7625.5 Subsets of ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7725.6 Related mathematical areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

25.6.1 Universal algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7725.6.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7725.6.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

25.7 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7825.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7825.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7825.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7825.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

26 Orthocompact space 8026.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

27 Paracompact space 8127.1 Paracompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8127.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8127.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8227.4 Paracompact Hausdor Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

27.4.1 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8327.5 Relationship with compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

27.5.1 Comparison of properties with compactness . . . . . . . . . . . . . . . . . . . . . . . . . 8427.6 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

27.6.1 Denition of relevant terms for the variations . . . . . . . . . . . . . . . . . . . . . . . . . 8527.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8527.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8527.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8627.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

28 Partially ordered set 8728.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8828.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8828.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8828.4 Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . 8928.5 Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8928.6 Strict and non-strict partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9028.7 Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

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28.8 Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9028.9 Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9128.10Linear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9128.11In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9228.12Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9228.13Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9228.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9228.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.17External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

29 Pseudocompact space 9429.1 Properties related to pseudocompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9429.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9429.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

30 Realcompact space 9630.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9630.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9630.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

31 Relatively compact subspace 9831.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9831.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

32 Sequentially compact space 9932.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9932.2 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9932.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9932.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9932.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

33 Set (mathematics) 10133.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10233.2 Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10233.3 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

33.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10433.3.2 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

33.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10533.5 Special sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10533.6 Basic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

33.6.1 Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10633.6.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

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33.6.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10733.6.4 Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

33.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11033.8 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11033.9 Principle of inclusion and exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11133.10De Morgans Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11133.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11233.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11233.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11233.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

34 Strictly singular operator 11334.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

35 Subset 11435.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11535.2 and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11535.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11535.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11635.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11635.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11635.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

36 Supercompact space 11836.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11836.2 Some Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11836.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

37 Topological space 12037.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

37.1.1 Neighbourhoods denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12037.1.2 Open sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12137.1.3 Closed sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12237.1.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

37.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12237.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12237.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12337.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12437.6 Classication of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12437.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12437.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12437.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12437.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

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37.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12537.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12537.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

38 Topology 12738.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12838.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12938.3 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

38.3.1 Topologies on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13138.3.2 Continuous functions and homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 13238.3.3 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

38.4 Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13238.4.1 General topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13238.4.2 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13338.4.3 Dierential topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13338.4.4 Geometric topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13338.4.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

38.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13438.5.1 Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13438.5.2 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13438.5.3 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13438.5.4 Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

38.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13438.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13538.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13638.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

39 Total order 13739.1 Strict total order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13739.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13839.3 Further concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

39.3.1 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13839.3.2 Lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13839.3.3 Finite total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13939.3.4 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13939.3.5 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13939.3.6 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13939.3.7 Sums of orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

39.4 Orders on the Cartesian product of totally ordered sets . . . . . . . . . . . . . . . . . . . . . . . . 14039.5 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14039.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14039.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

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39.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

40 Totally bounded space 14240.1 Denition for a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14240.2 Denitions in other contexts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14240.3 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14340.4 Relationships with compactness and completeness . . . . . . . . . . . . . . . . . . . . . . . . . . 14340.5 Use of the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14440.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14440.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14440.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

41 -compact space 14541.1 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14541.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14541.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14641.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14641.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 147

41.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14741.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15141.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Chapter 1

a-paracompact space

In mathematics, in the eld of topology, a topological space is said to be a-paracompact if every open cover of thespace has a locally nite renement. In contrast to the denition of paracompactness, the renement is not requiredto be open.Every paracompact space is a-paracompact, and in regular spaces the two notions coincide.

1.1 References Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

1

Chapter 2

Binary relation

Relation (mathematics)" redirects here. For a more general notion of relation, see nitary relation. For a morecombinatorial viewpoint, see theory of relations. For other uses, see Relation Mathematics.

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is asubset of the Cartesian product A2 = A A. More generally, a binary relation between two sets A and B is a subsetof A B. The terms correspondence, dyadic relation and 2-place relation are synonyms for binary relation.An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which everyprime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). Inthis relation, for instance, the prime 2 is associated with numbers that include 4, 0, 6, 10, but not 1 or 9; and theprime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", anddivides in arithmetic, "is congruent to" in geometry, is adjacent to in graph theory, is orthogonal to in linearalgebra and many more. The concept of function is dened as a special kind of binary relation. Binary relations arealso heavily used in computer science.A binary relation is the special case n = 2 of an n-ary relation R A1 An, that is, a set of n-tuples where thejth component of each n-tuple is taken from the jth domain Aj of the relation. An example for a ternary relation onZZZ is lies between ... and ..., containing e.g. the triples (5,2,8), (5,8,2), and (4,9,7).In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. Thisextension is needed for, among other things, modeling the concepts of is an element of or is a subset of in settheory, without running into logical inconsistencies such as Russells paradox.

2.1 Formal denition

A binary relation R is usually dened as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), andG is a subset of the Cartesian product X Y. The sets X and Y are called the domain (or the set of departure) andcodomain (or the set of destination), respectively, of the relation, and G is called its graph.The statement (x,y) G is read "x is R-related to y", and is denoted by xRy or R(x,y). The latter notation correspondsto viewing R as the characteristic function on X Y for the set of pairs of G.The order of the elements in each pair ofG is important: if a b, then aRb and bRa can be true or false, independentlyof each other. Resuming the above example, the prime 3 divides the integer 9, but 9 doesn't divide 3.A relation as dened by the triple (X, Y, G) is sometimes referred to as a correspondence instead.[1] In this case therelation from X to Y is the subset G of X Y, and from X to Y" must always be either specied or implied by thecontext when referring to the relation. In practice correspondence and relation tend to be used interchangeably.

2

2.2. SPECIAL TYPES OF BINARY RELATIONS 3

2.1.1 Is a relation more than its graph?According to the denition above, two relations with identical graphs but dierent domains or dierent codomainsare considered dierent. For example, ifG = f(1; 2); (1; 3); (2; 7)g , then (Z;Z; G) , (R;N; G) , and (N;R; G) arethree distinct relations, where Z is the set of integers and R is the set of real numbers.Especially in set theory, binary relations are often dened as sets of ordered pairs, identifying binary relations withtheir graphs. The domain of a binary relation R is then dened as the set of all x such that there exists at least oney such that (x; y) 2 R , the range of R is dened as the set of all y such that there exists at least one x such that(x; y) 2 R , and the eld of R is the union of its domain and its range.[2][3][4]A special case of this dierence in points of view applies to the notion of function. Many authors insist on distin-guishing between a functions codomain and its range. Thus, a single rule, like mapping every real number x tox2, can lead to distinct functions f : R ! R and f : R ! R+ , depending on whether the images under thatrule are understood to be reals or, more restrictively, non-negative reals. But others view functions as simply sets ofordered pairs with unique rst components. This dierence in perspectives does raise some nontrivial issues. As anexample, the former camp considers surjectivityor being ontoas a property of functions, while the latter sees itas a relationship that functions may bear to sets.Either approach is adequate for most uses, provided that one attends to the necessary changes in language, notation,and the denitions of concepts like restrictions, composition, inverse relation, and so on. The choice between the twodenitions usually matters only in very formal contexts, like category theory.

2.1.2 ExampleExample: Suppose there are four objects {ball, car, doll, gun} and four persons {John, Mary, Ian, Venus}. Supposethat John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the gun and Ian owns nothing.Then the binary relation is owned by is given as

R = ({ball, car, doll, gun}, {John, Mary, Ian, Venus}, {(ball, John), (doll, Mary), (car, Venus)}).

Thus the rst element of R is the set of objects, the second is the set of persons, and the last element is a set of orderedpairs of the form (object, owner).The pair (ball, John), denoted by RJ means that the ball is owned by John.Two dierent relations could have the same graph. For example: the relation

({ball, car, doll, gun}, {John, Mary, Venus}, {(ball, John), (doll, Mary), (car, Venus)})

is dierent from the previous one as everyone is an owner. But the graphs of the two relations are the same.Nevertheless, R is usually identied or even dened as G(R) and an ordered pair (x, y) G(R)" is usually denoted as"(x, y) R".

2.2 Special types of binary relationsSome important types of binary relations R between two sets X and Y are listed below. To emphasize that X and Ycan be dierent sets, some authors call such binary relations heterogeneous.[5][6]

Uniqueness properties:

injective (also called left-unique[7]): for all x and z in X and y in Y it holds that if xRy and zRy then x = z. Forexample, the green relation in the diagram is injective, but the red relation is not, as it relates e.g. both x = 5and z = +5 to y = 25.

functional (also called univalent[8] or right-unique[7] or right-denite[9]): for all x in X, and y and z in Yit holds that if xRy and xRz then y = z; such a binary relation is called a partial function. Both relations inthe picture are functional. An example for a non-functional relation can be obtained by rotating the red graphclockwise by 90 degrees, i.e. by considering the relation x=y2 which relates e.g. x=25 to both y=5 and z=+5.

4 CHAPTER 2. BINARY RELATION

Example relations between real numbers. Red: y=x2. Green: y=2x+20.

one-to-one (also written 1-to-1): injective and functional. The green relation is one-to-one, but the red is not.

Totality properties:

left-total:[7] for all x in X there exists a y in Y such that xRy. For example R is left-total when it is a functionor a multivalued function. Note that this property, although sometimes also referred to as total, is dierentfrom the denition of total in the next section. Both relations in the picture are left-total. The relation x=y2,obtained from the above rotation, is not left-total, as it doesn't relate, e.g., x = 14 to any real number y.

surjective (also called right-total[7] or onto): for all y in Y there exists an x in X such that xRy. The greenrelation is surjective, but the red relation is not, as it doesn't relate any real number x to e.g. y = 14.

Uniqueness and totality properties:

2.3. RELATIONS OVER A SET 5

A function: a relation that is functional and left-total. Both the green and the red relation are functions. An injective function: a relation that is injective, functional, and left-total. A surjective function or surjection: a relation that is functional, left-total, and right-total. A bijection: a surjective one-to-one or surjective injective function is said to be bijective, also known asone-to-one correspondence.[10] The green relation is bijective, but the red is not.

2.2.1 DifunctionalLess commonly encountered is the notion of difunctional (or regular) relation, dened as a relation R such thatR=RR1R.[11]

To understand this notion better, it helps to consider a relation as mapping every element xX to a set xR = { yY| xRy }.[11] This set is sometimes called the successor neighborhood of x in R; one can dene the predecessorneighborhood analogously.[12] Synonymous terms for these notions are afterset and respectively foreset.[5]

A difunctional relation can then be equivalently characterized as a relation R such that wherever x1R and x2R have anon-empty intersection, then these two sets coincide; formally x1R x2R implies x1R = x2R.[11]

As examples, any function or any functional (right-unique) relation is difunctional; the converse doesn't hold. If oneconsiders a relation R from set to itself (X = Y), then if R is both transitive and symmetric (i.e. a partial equivalencerelation), then it is also difunctional.[13] The converse of this latter statement also doesn't hold.A characterization of difunctional relations, which also explains their name, is to consider two functions f: A Cand g: B C and then dene the following set which generalizes the kernel of a single function as joint kernel: ker(f,g) = { (a, b) A B | f(a) = g(b) }. Every difunctional relation R A B arises as the joint kernel of two functionsf: A C and g: B C for some set C.[14]

In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This ter-minology is justied by the fact that when represented as a boolean matrix, the columns and rows of a difunctionalrelation can be arranged in such a way as to present rectangular blocks of true on the (asymmetric) main diagonal.[15]Other authors however use the term rectangular to denote any heterogeneous relation whatsoever.[6]

2.3 Relations over a setIf X = Y then we simply say that the binary relation is over X, or that it is an endorelation over X.[16] In computerscience, such a relation is also called a homogeneous (binary) relation.[16][17][6] Some types of endorelations arewidely studied in graph theory, where they are known as simple directed graphs permitting loops.The set of all binary relations Rel(X) on a set X is the set 2X X which is a Boolean algebra augmented with theinvolution of mapping of a relation to its inverse relation. For the theoretical explanation see Relation algebra.Some important properties of a binary relation R over a set X are:

reexive: for all x in X it holds that xRx. For example, greater than or equal to () is a reexive relation butgreater than (>) is not.

irreexive (or strict): for all x in X it holds that not xRx. For example, > is an irreexive relation, but is not. coreexive: for all x and y in X it holds that if xRy then x = y. An example of a coreexive relation is therelation on integers in which each odd number is related to itself and there are no other relations. The equalityrelation is the only example of a both reexive and coreexive relation.

The previous 3 alternatives are far from being exhaustive; e.g. the red relation y=x2 from theabove picture is neither irreexive, nor coreexive, nor reexive, since it contains the pair(0,0), and (2,4), but not (2,2), respectively.

symmetric: for all x and y in X it holds that if xRy then yRx. Is a blood relative of is a symmetric relation,because x is a blood relative of y if and only if y is a blood relative of x.


antisymmetric: for all x and y in X, if xRy and yRx then x = y. For example, is anti-symmetric (so is >, butonly because the condition in the denition is always false).[18]

asymmetric: for all x and y in X, if xRy then not yRx. A relation is asymmetric if and only if it is bothanti-symmetric and irreexive.[19] For example, > is asymmetric, but is not.

transitive: for all x, y and z in X it holds that if xRy and yRz then xRz. A transitive relation is irreexive if andonly if it is asymmetric.[20] For example, is ancestor of is transitive, while is parent of is not.

total: for all x and y in X it holds that xRy or yRx (or both). This denition for total is dierent from left totalin the previous section. For example, is a total relation.

trichotomous: for all x and y in X exactly one of xRy, yRx or x = y holds. For example, > is a trichotomousrelation, while the relation divides on natural numbers is not.[21]

Euclidean: for all x, y and z in X it holds that if xRy and xRz, then yRz (and zRy). Equality is a Euclideanrelation because if x=y and x=z, then y=z.

serial: for all x in X, there exists y in X such that xRy. "Is greater than" is a serial relation on the integers. Butit is not a serial relation on the positive integers, because there is no y in the positive integers (i.e. the naturalnumbers) such that 1>y.[22] However, "is less than" is a serial relation on the positive integers, the rationalnumbers and the real numbers. Every reexive relation is serial: for a given x, choose y=x. A serial relation canbe equivalently characterized as every element having a non-empty successor neighborhood (see the previoussection for the denition of this notion). Similarly an inverse serial relation is a relation in which every elementhas non-empty predecessor neighborhood.[12]

set-like (or local): for every x in X, the class of all y such that yRx is a set. (This makes sense only if relationson proper classes are allowed.) The usual ordering < on the class of ordinal numbers is set-like, while its inverse> is not.

A relation that is reexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric,transitive, and serial is also reexive. A relation that is only symmetric and transitive (without necessarily beingreexive) is called a partial equivalence relation.A relation that is reexive, antisymmetric, and transitive is called a partial order. A partial order that is total is calleda total order, simple order, linear order, or a chain.[23] A linear order where every nonempty subset has a least elementis called a well-order.

2.4 Operations on binary relationsIf R, S are binary relations over X and Y, then each of the following is a binary relation over X and Y :

Union: R S X Y, dened as R S = { (x, y) | (x, y) R or (x, y) S }. For example, is the union of >and =.

Intersection: R S X Y, dened as R S = { (x, y) | (x, y) R and (x, y) S }.

If R is a binary relation over X and Y, and S is a binary relation over Y and Z, then the following is a binary relationover X and Z: (see main article composition of relations)

Composition: S R, also denoted R ; S (or more ambiguously R S), dened as S R = { (x, z) | there existsy Y, such that (x, y) R and (y, z) S }. The order of R and S in the notation S R, used here agrees withthe standard notational order for composition of functions. For example, the composition is mother of isparent of yields is maternal grandparent of, while the composition is parent of is mother of yields isgrandmother of.

2.4. OPERATIONS ON BINARY RELATIONS 7

A relation R on sets X and Y is said to be contained in a relation S on X and Y if R is a subset of S, that is, if x R yalways implies x S y. In this case, if R and S disagree, R is also said to be smaller than S. For example, > is containedin .If R is a binary relation over X and Y, then the following is a binary relation over Y and X:

Inverse or converse: R 1, dened as R 1 = { (y, x) | (x, y) R }. A binary relation over a set is equal to itsinverse if and only if it is symmetric. See also duality (order theory). For example, is less than ().

If R is a binary relation over X, then each of the following is a binary relation over X:

Reexive closure: R =, dened as R = = { (x, x) | x X } R or the smallest reexive relation over X containingR. This can be proven to be equal to the intersection of all reexive relations containing R.

Reexive reduction: R , dened as R = R \ { (x, x) | x X } or the largest irreexive relation over Xcontained in R.

Transitive closure: R +, dened as the smallest transitive relation over X containing R. This can be seen to beequal to the intersection of all transitive relations containing R.

Transitive reduction: R , dened as a minimal relation having the same transitive closure as R. Reexive transitive closure: R *, dened as R * = (R +) =, the smallest preorder containing R. Reexive transitive symmetric closure: R , dened as the smallest equivalence relation over X containingR.

2.4.1 ComplementIf R is a binary relation over X and Y, then the following too:

The complement S is dened as x S y if not x R y. For example, on real numbers, is the complement of >.

The complement of the inverse is the inverse of the complement.If X = Y, the complement has the following properties:

If a relation is symmetric, the complement is too. The complement of a reexive relation is irreexive and vice versa. The complement of a strict weak order is a total preorder and vice versa.

The complement of the inverse has these same properties.

2.4.2 RestrictionThe restriction of a binary relation on a set X to a subset S is the set of all pairs (x, y) in the relation for which x andy are in S.If a relation is reexive, irreexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partialorder, total order, strict weak order, total preorder (weak order), or an equivalence relation, its restrictions are too.However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in generalnot equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother ofthe woman y"; its transitive closure doesn't relate a woman with her paternal grandmother. On the other hand, thetransitive closure of is parent of is is ancestor of"; its restriction to females does relate a woman with her paternalgrandmother.


Also, the various concepts of completeness (not to be confused with being total) do not carry over to restrictions.For example, on the set of real numbers a property of the relation "" is that every non-empty subset S of R with anupper bound in R has a least upper bound (also called supremum) in R. However, for a set of rational numbers thissupremum is not necessarily rational, so the same property does not hold on the restriction of the relation "" to theset of rational numbers.The left-restriction (right-restriction, respectively) of a binary relation between X and Y to a subset S of its domain(codomain) is the set of all pairs (x, y) in the relation for which x (y) is an element of S.

2.4.3 Algebras, categories, and rewriting systemsVarious operations on binary endorelations can be treated as giving rise to an algebraic structure, known as relationalgebra. It should not be confused with relational algebra which deals in nitary relations (and in practice also niteand many-sorted).For heterogenous binary relations, a category of relations arises.[6]

Despite their simplicity, binary relations are at the core of an abstract computation model known as an abstractrewriting system.

2.5 Sets versus classesCertain mathematical relations, such as equal to, member of, and subset of, cannot be understood to be binaryrelations as dened above, because their domains and codomains cannot be taken to be sets in the usual systems ofaxiomatic set theory. For example, if we try to model the general concept of equality as a binary relation =, wemust take the domain and codomain to be the class of all sets, which is not a set in the usual set theory.In most mathematical contexts, references to the relations of equality, membership and subset are harmless becausethey can be understood implicitly to be restricted to some set in the context. The usual work-around to this problemis to select a large enough set A, that contains all the objects of interest, and work with the restriction =A instead of=. Similarly, the subset of relation needs to be restricted to have domain and codomain P(A) (the power set ofa specic set A): the resulting set relation can be denoted A. Also, the member of relation needs to be restrictedto have domain A and codomain P(A) to obtain a binary relation A that is a set. Bertrand Russell has shown thatassuming to be dened on all sets leads to a contradiction in naive set theory.Another solution to this problem is to use a set theory with proper classes, such as NBG or MorseKelley set theory,and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership,and subset are binary relations without special comment. (A minor modication needs to be made to the concept ofthe ordered triple (X, Y, G), as normally a proper class cannot be a member of an ordered tuple; or of course onecan identify the function with its graph in this context.)[24] With this denition one can for instance dene a functionrelation between every set and its power set.

2.6 The number of binary relations

The number of distinct binary relations on an n-element set is 2n2 (sequence A002416 in OEIS):Notes:

The number of irreexive relations is the same as that of reexive relations. The number of strict partial orders (irreexive transitive relations) is the same as that of partial orders. The number of strict weak orders is the same as that of total preorders. The total orders are the partial orders that are also total preorders. The number of preorders that are neithera partial order nor a total preorder is, therefore, the number of preorders, minus the number of partial orders,minus the number of total preorders, plus the number of total orders: 0, 0, 0, 3, and 85, respectively.

the number of equivalence relations is the number of partitions, which is the Bell number.

2.7. EXAMPLES OF COMMON BINARY RELATIONS 9

The binary relations can be grouped into pairs (relation, complement), except that for n = 0 the relation is its owncomplement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse com-plement).

2.7 Examples of common binary relations order relations, including strict orders:

greater than greater than or equal to less than less than or equal to divides (evenly) is a subset of

equivalence relations: equality is parallel to (for ane spaces) is in bijection with isomorphy

dependency relation, a nite, symmetric, reexive relation. independency relation, a symmetric, irreexive relation which is the complement of some dependency relation.

2.8 See also Conuence (term rewriting) Hasse diagram Incidence structure Logic of relatives Order theory Triadic relation

2.9 Notes[1] Encyclopedic dictionary of Mathematics. MIT. 2000. pp. 13301331. ISBN 0-262-59020-4.

[2] Suppes, Patrick (1972) [originally published by D. van Nostrand Company in 1960]. Axiomatic Set Theory. Dover. ISBN0-486-61630-4.

[3] Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7.

[4] Levy, Azriel (2002) [republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979].Basic Set Theory. Dover. ISBN 0-486-42079-5.

[5] Christodoulos A. Floudas; PanosM. Pardalos (2008). Encyclopedia of Optimization (2nd ed.). Springer Science&BusinessMedia. pp. 299300. ISBN 978-0-387-74758-3.


[6] Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. xxi. ISBN978-1-4020-6164-6.

[7] Kilp, Knauer and Mikhalev: p. 3. The same four denitions appear in the following:

Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook.Springer Science & Business Media. p. 506. ISBN 978-3-540-67995-0.

Eike Best (1996). Semantics of Sequential and Parallel Programs. Prentice Hall. pp. 1921. ISBN 978-0-13-460643-9.

Robert-Christoph Riemann (1999). Modelling of Concurrent Systems: Structural and Semantical Methods in the HighLevel Petri Net Calculus. Herbert Utz Verlag. pp. 2122. ISBN 978-3-89675-629-9.

[8] Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7, Chapt. 5

[9] Ms, Stephan (2007), Reasoning on Spatial Semantic Integrity Constraints, Spatial Information Theory: 8th InternationalConference, COSIT 2007, Melbourne, Australiia, September 1923, 2007, Proceedings, Lecture Notes in Computer Science4736, Springer, pp. 285302, doi:10.1007/978-3-540-74788-8_18

[10] Note that the use of correspondence here is narrower than as general synonym for binary relation.

[11] Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer Science &Business Media. p. 200. ISBN 978-3-211-82971-4.

[12] Yao, Y. (2004). Semantics of Fuzzy Sets in Rough Set Theory. Transactions on Rough Sets II. Lecture Notes in ComputerScience 3135. p. 297. doi:10.1007/978-3-540-27778-1_15. ISBN 978-3-540-23990-1.

[13] William Craig (2006). Semigroups Underlying First-order Logic. American Mathematical Soc. p. 72. ISBN 978-0-8218-6588-0.

[14] Gumm, H. P.; Zarrad, M. (2014). Coalgebraic Simulations and Congruences. Coalgebraic Methods in Computer Science.Lecture Notes in Computer Science 8446. p. 118. doi:10.1007/978-3-662-44124-4_7. ISBN 978-3-662-44123-7.

[15] Julius Richard Bchi (1989). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions.Springer Science & Business Media. pp. 3537. ISBN 978-1-4613-8853-1.

[16] M. E. Mller (2012). Relational Knowledge Discovery. Cambridge University Press. p. 22. ISBN 978-0-521-19021-3.

[17] Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook. SpringerScience & Business Media. p. 496. ISBN 978-3-540-67995-0.

[18] Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006),ATransition to AdvancedMathematics (6th ed.), Brooks/Cole,p. 160, ISBN 0-534-39900-2

[19] Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography,Springer-Verlag, p. 158.

[20] Flaka, V.; Jeek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics Physics Charles University. p. 1. Lemma 1.1 (iv). This source refers to asymmetric relations as strictlyantisymmetric.

[21] Since neither 5 divides 3, nor 3 divides 5, nor 3=5.

[22] Yao, Y.Y.; Wong, S.K.M. (1995). Generalization of rough sets using relationships between attribute values (PDF).Proceedings of the 2nd Annual Joint Conference on Information Sciences: 3033..

[23] Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, ISBN 0-12-597680-1, p. 4

[24] Tarski, Alfred; Givant, Steven (1987). A formalization of set theory without variables. American Mathematical Society. p.3. ISBN 0-8218-1041-3.

2.10 References M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories: with Applications to Wreath Products andGraphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.

Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.

2.11. EXTERNAL LINKS 11

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

Chapter 3

Closure (topology)

For other uses, see Closure (disambiguation).

In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S.The closure of S is also dened as the union of S and its boundary. Intuitively, these are all the points in S and nearS. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to thenotion of interior.

3.1 Denitions

3.1.1 Point of closureFor S a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S(this point may be x itself).This denition generalises to any subset S of a metric space X. Fully expressed, for X a metric space with metric d, xis a point of closure of S if for every r > 0, there is a y in S such that the distance d(x, y) < r. (Again, we may have x= y.) Another way to express this is to say that x is a point of closure of S if the distance d(x, S) := inf{d(x, s) : s inS} = 0.This denition generalises to topological spaces by replacing open ball or ball with "neighbourhood". Let S bea subset of a topological space X. Then x is a point of closure (or adherent point) of S if every neighbourhood of xcontains a point of S.[1] Note that this denition does not depend upon whether neighbourhoods are required to beopen.

3.1.2 Limit pointThe denition of a point of closure is closely related to the denition of a limit point. The dierence between thetwo denitions is subtle but important namely, in the denition of limit point, every neighborhood of the point xin question must contain a point of the set other than x itself.Thus, every limit point is a point of closure, but not every point of closure is a limit point. A point of closure whichis not a limit point is an isolated point. In other words, a point x is an isolated point of S if it is an element of S andif there is a neighbourhood of x which contains no other points of S other than x itself.[2]

For a given set S and point x, x is a point of closure of S if and only if x is an element of S or x is a limit point of S(or both).

3.1.3 Closure of a setSee also: Closure (mathematics)

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3.2. EXAMPLES 13

The closure of a set S is the set of all points of closure of S, that is, the set S together with all of its limit points.[3]The closure of S is denoted cl(S), Cl(S), S or S . The closure of a set has the following properties.[4]

cl(S) is a closed superset of S. cl(S) is the intersection of all closed sets containing S. cl(S) is the smallest closed set containing S. cl(S) is the union of S and its boundary (S). A set S is closed if and only if S = cl(S). If S is a subset of T, then cl(S) is a subset of cl(T). If A is a closed set, then A contains S if and only if A contains cl(S).

Sometimes the second or third property above is taken as the denition of the topological closure, which still makesense when applied to other types of closures (see below).[5]

In a rst-countable space (such as a metric space), cl(S) is the set of all limits of all convergent sequences of pointsin S. For a general topological space, this statement remains true if one replaces sequence by "net" or "lter".Note that these properties are also satised if closure, superset, intersection, contains/containing, smallestand closed are replaced by interior, subset, union, contained in, largest, and open. For more on thismatter, see closure operator below.

3.2 ExamplesConsider a sphere in 3 dimensions. Implicitly there are two regions of interest created by this sphere; the sphere itselfand its interior (which is called an open 3-ball). It is useful to be able to distinguish between the interior of 3-ball andthe surface, so we distinguish between the open 3-ball, and the closed 3-ball - the closure of the 3-ball. The closureof the open 3-ball is the open 3-ball plus the surface.In topological space:

In any space, ? = cl(?) . In any space X, X = cl(X).

Giving R and C the standard (metric) topology:

If X is the Euclidean space R of real numbers, then cl((0, 1)) = [0, 1]. If X is the Euclidean space R, then the closure of the set Q of rational numbers is the whole space R. We saythat Q is dense in R.

If X is the complex plane C = R2, then cl({z in C : |z| > 1}) = {z in C : |z| 1}. If S is a nite subset of a Euclidean space, then cl(S) = S. (For a general topological space, this property isequivalent to the T1 axiom.)

On the set of real numbers one can put other topologies rather than the standard one.

If X = R, where R has the lower limit topology, then cl((0, 1)) = [0, 1). If one considers on R the discrete topology in which every set is closed (open), then cl((0, 1)) = (0, 1). If one considers on R the trivial topology in which the only closed (open) sets are the empty set and R itself,then cl((0, 1)) = R.

14 CHAPTER 3. CLOSURE (TOPOLOGY)

These examples show that the closure of a set depends upon the topology of the underlying space. The last twoexamples are special cases of the following.

In any discrete space, since every set is closed (and also open), every set is equal to its closure. In any indiscrete space X, since the only closed sets are the empty set and X itself, we have that the closureof the empty set is the empty set, and for every non-empty subset A of X, cl(A) = X. In other words, everynon-empty subset of an indiscrete space is dense.

The closure of a set also depends upon in which space we are taking the closure. For example, if X is the set ofrational numbers, with the usual relative topology induced by the Euclidean space R, and if S = {q in Q : q2 > 2, q >0}, then S is closed in Q, and the closure of S in Q is S; however, the closure of S in the Euclidean space R is the setof all real numbers greater than or equal to

p2:

3.3 Closure operatorSee also: Closure operator

A closure operator on a set X is a mapping of the power set of X, P(X) , into itself which satises the Kuratowskiclosure axioms.Given a topological space (X; T ) , the mapping : S S for all S X is a closure operator on X. Conversely, if cis a closure operator on a set X, a topological space is obtained by dening the sets S with c(S) = S as closed sets (sotheir complements are the open sets of the topology).[6]

The closure operator is dual to the interior operator o, in the sense that

S = X \ (X \ S)o

and also

So = X \ (X \ S)

where X denotes the underlying set of the topological space containing S, and the backslash refers to the set-theoreticdierence.Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be easily translated intothe language of interior operators, by replacing sets with their complements.

3.4 Facts about closuresThe set S is closed if and only if Cl(S) = S . In particular:

The closure of the empty set is the empty set; The closure of X itself is X . The closure of an intersection of sets is always a subset of (but need not be equal to) the intersection of theclosures of the sets.

In a union of nitely many sets, the closure of the union and the union of the closures are equal; the union ofzero sets is the empty set, and so this statement contains the earlier statement about the closure of the emptyset as a special case.

The closure of the union of innitely many sets need not equal the union of the closures, but it is always asuperset of the union of the closures.

If A is a subspace ofX containing S , then the closure of S computed in A is equal to the intersection of A and theclosure of S computed inX : ClA(S) = A \ClX(S) . In particular, S is dense in A if and only if A is a subset ofClX(S) .

3.5. CATEGORICAL INTERPRETATION 15

3.5 Categorical interpretationOne may elegantly dene the closure operator in terms of universal arrows, as follows.The powerset of a set X may be realized as a partial order category P in which the objects are subsets and themorphisms are inclusions A ! B whenever A is a subset of B. Furthermore, a topology T on X is a subcategory ofP with inclusion functor I : T ! P . The set of closed subsets containing a xed subset A X can be identiedwith the comma category (A # I) . This category also a partial order then has initial object Cl(A). Thus thereis a universal arrow from A to I, given by the inclusion A! Cl(A) .Similarly, since every closed set containing X \ A corresponds with an open set contained in A we can interpret thecategory (I # X nA) as the set of open subsets contained in A, with terminal object int(A) , the interior of A.All properties of the closure can be derived from this denition and a few properties of the above categories. More-over, this denition makes precise the analogy between the topological closure and other types of closures (forexample algebraic), since all are examples of universal arrows.

3.6 See also Closure algebra

3.7 Notes[1] Schubert, p. 20

[2] Kuratowski, p. 75

[3] Hocking Young, p. 4

[4] Croom, p. 104

[5] Gemignani, p. 55, Pervin, p. 40 and Baker, p. 38 use the second property as the denition.

[6] Pervin, p. 41

3.8 References Baker, Crump W. (1991), Introduction to Topology, Wm. C. Brown Publisher, ISBN 0-697-05972-3 Croom, Fred H. (1989), Principles of Topology, Saunders College Publishing, ISBN 0-03-012813-7 Gemignani, Michael C. (1990) [1967], Elementary Topology (2nd ed.), Dover, ISBN 0-486-66522-4 Hocking, John G.; Young, Gail S. (1988) [1961], Topology, Dover, ISBN 0-486-65676-4 Kuratowski, K. (1966), Topology I, Academic Press Pervin, William J. (1965), Foundations of General Topology, Academic Press Schubert, Horst (1968), Topology, Allyn and Bacon

3.9 External links Hazewinkel, Michiel, ed. (2001), Closure of a set, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 4

Compact operator

In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space Xto another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset ofY. Such an operator is necessarily a bounded operator, and so continuous.Any bounded operator L that has nite rank is a compact operator; indeed, the class of compact operators is a naturalgeneralisation of the class of nite-rank operators in an innite-dimensional setting. When Y is a Hilbert space, itis true that any compact operator is a limit of nite-rank operators, so that the class of compact operators can bedened alternatively as the closure in the operator norm of the nite-rank operators. Whether this was true in generalfor Banach spaces (the approximation property) was an unsolved question for many years; in the end Per Eno gavea counter-example.The origin of the theory of compact operators is in the theory of integral equations, where integral operators supplyconcrete examples of such operators. A typical Fredholm integral equation gives rise to a compact operator K onfunction spaces; the compactness property is shown by equicontinuity. The method of approximation by nite-rankoperators is basic in the numerical solution of such equations. The abstract idea of Fredholm operator is derived fromthis connection.

4.1 Equivalent formulationsA bounded operator T : X Y is compact if and only if any of the following is true

Image of the unit ball in X under T is relatively compact in Y. Image of any bounded set under T is relatively compact in Y. Image of any bounded set under T is totally bounded in Y. there exists a neighbourhood of 0, U X , and compact set V Y such that T (U) V . For any sequence (xn)n2N from the unit ball in X, the sequence (Txn)n2N contains a Cauchy subsequence.

Note that if a linear operator is compact, then it is easy to see that it is bounded, and hence continuous.

4.2 Important propertiesIn the following, X, Y, Z, W are Banach spaces, B(X, Y) is the space of bounded operators from X to Y with theoperator norm, K(X, Y) is the space of compact operators from X to Y, B(X) = B(X, X), K(X) = K(X, X), idX is theidentity operator on X.

K(X, Y) is a closed subspace of B(X, Y): Let Tn, n N, be a sequence of compact operators from one Banachspace to the other, and suppose that Tn converges to T with respect to the operator norm. Then T is alsocompact.

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4.3. ORIGINS IN INTEGRAL EQUATION THEORY 17

Conversely, if X, Y are Hilbert spaces, then every compact operator from X to Y is the limit of nite rankoperators. Notably, this is false for general Banach spaces X and Y.

B(Y;Z) K(X;Y ) B(W;X) K(W;Z): In particular, K(X) forms a two-sided operator ideal in B(X).

idX is compact if and only if X has nite dimension.

For any T K(X), idX T is a Fredholm operator of index 0. In particular, im (idX T ) is closed. Thisis essential in developing the spectral properties of compact operators. One can notice the similarity betweenthis property and the fact that, if M and N are subspaces of a Banach space where M is closed and N isnite-dimensional, then M + N is also closed.

Any compact operator is strictly singular, but not vice versa.[1]

An operator is compact if and only if its adjoint is compact (Schauders theorem).

4.3 Origins in integral equation theoryA crucial property of compact operators is the Fredholm alternative, which asserts that the existence of solution oflinear equations of the form(K + I)u = f

(where K is a compact operator, f is a given function, and u is the unknown function to be solved for) behaves muchlike as in nite dimensions. The spectral theory of compact operators then follows, and it is due to Frigyes Riesz(1918). It shows that a compact operator K on an innite-dimensional Banach space has spectrum that is either anite subset of C which includes 0, or the spectrum is a countably innite subset of C which has 0 as its only limitpoint. Moreover, in either case the non-zero elements of the spectrum are eigenvalues of K with nite multiplicities(so that K I has a nite-dimensional kernel for all complex 0).An important example of a compact operator is compact embedding of Sobolev spaces, which, along with the Grdinginequality and the LaxMilgram theorem, can be used to convert an elliptic boundary value problem into a Fredholmintegral equation.[2] Existence of the solution and spectral properties then follow from the theory of compact opera-tors; in particular, an elliptic boundary value problem on a bounded domain has innitely many isolated eigenvalues.One consequence is that a solid body can vibrate only at isolated frequencies, given by the eigenvalues, and arbitrarilyhigh vibration frequencies always exist.The compact operators from a Banach space to itself form a two-sided ideal in the algebra of all bounded operatorson the space. Indeed, the compact operators on an innite-dimensional Hilbert space form a maximal ideal, so thequotient algebra, known as the Calkin algebra, is simple.

4.4 Compact operator on Hilbert spacesMain article: Compact operator on Hilbert space

An equivalent denition of compact operators on a Hilbert space may be given as follows.An operator T on an innite-dimensional Hilbert spaceH

T : H ! His said to be compact if it can be written in the form

T =1Xn=1

nhfn; ign ;

18 CHAPTER 4. COMPACT OPERATOR

where f1; f2; : : : and g1; g2; : : : are (not necessarily complete) orthonormal sets, and 1; 2; : : : is a sequence ofpositive numbers with limit zero, called the singular values of the operator. The singular values can accumulate onlyat zero. If the sequence becomes stationary at zero, that is N+k = 0 for some N 2 N; and every k = 1; 2; : : : ,then the operator has nite rank, i.e, a nite-dimensional range and can be written as

T =NXn=1

nhfn; ign :

The bracket h; i is the scalar product on the Hilbert space; the sum on the right hand side converges in the operatornorm.An important subclass of compact operators is the trace-class or nuclear operators.

4.5 Completely continuous operatorsLet X and Y be Banach spaces. A bounded linear operator T : X Y is called completely continuous if, for everyweakly convergent sequence (xn) from X, the sequence (Txn) is norm-convergent in Y (Conway 1985, VI.3).Compact operators on a Banach space are always completely continuous. If X is a reexive Banach space, then everycompletely continuous operator T : X Y is compact.

4.6 Examples Every nite rank operator is compact.

For `p and a sequence (tn) converging to zero, the multiplication operator (Tx)n = tn xn is compact.

For some xed g C([0, 1]; R), dene the linear operator T from C([0, 1]; R) to C([0, 1]; R) by

(Tf)(x) =

Z x0

f(t)g(t) dt:

That the operator T is indeed compact follows from the Ascoli theorem.

More generally, if is any domain in Rn and the integral kernel k : R is a HilbertSchmidt kernel,then the operator T on L2(; R) dened by

(Tf)(x) =

Z

k(x; y)f(y) dy

is a compact operator.

By Rieszs lemma, the identity operator is a compact operator if and only if the space is nite-dimensional.

4.7 See also Spectral theory of compact operators Fredholm operator Fredholm integral equations Fredholm alternative Compact embedding Strictly singular operator

4.8. NOTES 19

4.8 Notes[1] N.L. Carothers, A Short Course on Banach Space Theory, (2005) London Mathematical Society Student Texts 64, Cam-

bridge University Press.

[2] William McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000

4.9 References Conway, John B. (1985). A course in functional analysis. Springer-Verlag. ISBN 3-540-96042-2.

Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial dierential equations. Texts inApplied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. 356. ISBN 0-387-00444-0. (Section7.5)

Kutateladze, S.S. (1996). Fundamentals of Functional Analysis. Texts in Mathematical Sciences 12 (Seconded.). New York: Springer-Verlag. p. 292. ISBN 978-0-7923-3898-7.

Chapter 5

Compact space

Compactness redirects here. For the concept in rst-order logic, see Compactness theorem.In mathematics, and more specically in general topology, compactness is a property that generalizes the notion of

The interval A = (-, 2] is not compact because it is not bounded. The interval C = (2, 4) is not compact because it is not closed.The interval B = [0, 1] is compact because it is both closed and bounded.

a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all itspoints lie within some xed distance of each other). Examples include a closed interval, a rectangle, or a nite set ofpoints. This notion is dened for more general topological spaces than Euclidean space in various ways.One such generalization is that a space is sequentially compact if any innite sequence of points sampled from thespace must frequently (innitely often) get arbitrarily close to some point of the space. An equivalent denition isthat every sequence of points must have an innite subsequence that converges to some point of the space. TheHeine-Borel theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it isclosed and bounded. Thus, if one chooses an innite number of points in the closed unit interval [0, 1] some of thosepoints must get arbitrarily close to some real number in that space. For instance, some of the numbers 1/2, 4/5, 1/3,5/6, 1/4, 6/7, accumulate to 0 (others accumulate to 1). The same set of points would not accumulate to any pointof the open unit interval (0, 1); so the open unit interval is not compact. Euclidean space itself is not compact sinceit is not bounded. In particular, the sequence of points 0, 1, 2, 3, has no subsequence that converges to any givenreal number.Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spacesconsisting not of geometrical points but of functions. The term compact was introduced into mathematics by MauriceFrchet in 1904 as a distillation of this concept. Compactness in this more general situation plays an extremelyimportant role in mathematical analysis, because many classical and important theorems of 19th century analysis,such as the extreme value theorem, are easily generalized to this situation. A typical application is furnished by the

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5.1. HISTORICAL DEVELOPMENT 21

ArzelAscoli theorem or the Peano existence theorem, in which one is able to conclude the existence of a functionwith some required properties as a limiting case of some more elementary construction.Various equivalent notions of compactness, including sequential compactness and limit point compactness, can bedeveloped in general metric spaces. In general topological spaces, however, dierent notions of compactness are notnecessarily equivalent. The most useful notion, which is the standard denition of the unqualied term compactness,is phrased in terms of the existence of nite families of open sets that "cover" the space in the sense that each pointof the space must lie in some set contained in the family. This more subtle notion, introduced by Pavel Alexandrovand Pavel Urysohn in 1929, exhibits compact spaces as generalizations of nite sets. In spaces that are compact inthis sense, it is often possible to patch together information that holds locallythat is, in a neighborhood of eachpointinto corresponding statements that hold throughout the space, and many theorems are of this character.The term compact set is sometimes a synonym for compact space, but usually refers to a compact subspace of atopological space.

5.1 Historical developmentIn the 19th century, several disparate mathematical properties were understood that would later be seen as conse-quences of compactness. On the one hand, Bernard Bolzano (1817) had been aware that any bounded sequence ofpoints (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some otherpoint, called a limit point. Bolzanos proof relied on the method of bisection: the sequence was placed into an intervalthat was then divided into two equal parts, and a part containing innitely many terms of the sequence was selected.The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts until itcloses down on the desired limit point. The full signicance of Bolzanos theorem, and its method of proof, wouldnot emerge until almost 50 years later when it was rediscovered by Karl Weierstrass.[1]

In the 1880s, it became clear that results similar to the BolzanoWeierstrass theorem could be formulated for spacesof functions rather than just numbers or geometrical points. The idea of regarding functions as themselves pointsof a generalized space dates back to the investigations of Giulio Ascoli and Cesare Arzel.[2] The culmination oftheir investigations, the ArzelAscoli theorem, was a generalization of the BolzanoWeierstrass theorem to familiesof continuous functions, the precise conclusion of which was that it was possible to extract a uniformly convergentsequence of functions from a suitable family of functions. The uniform limit of this sequence then played preciselythe same role as Bolzanos limit point. Towards the beginning of the twentieth century, results similar to that ofArzel and Ascoli began to accumulate in the area of integral equations, as investigated by David Hilbert and ErhardSchmidt. For a certain class of Green functions coming from solutions of integral equations, Schmidt had shown thata property analogous to the ArzelAscoli theorem held in the sense of mean convergenceor convergence in whatwould later be dubbed a Hilbert space. This ultimately led to the notion of a compac

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