Axiom of countabilityWikipedia

Contents

1 Axiom of countability 11.1 Important examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Relationships with each other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Ball (mathematics) 32.1 Balls in Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 The volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Balls in general metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Balls in normed vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.1 p-norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.2 General convex norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Topological balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Base (topology) 73.1 Simple properties of bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Objects dened in terms of bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 Base for the closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.5 Weight and character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.5.1 Increasing chains of open sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Category (mathematics) 114.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3 Small and large categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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4.5 Construction of new categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.5.1 Dual category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.5.2 Product categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.6 Types of morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.7 Types of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Closure (topology) 175.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.1.1 Point of closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.1.2 Limit point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.1.3 Closure of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3 Closure operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.4 Facts about closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.5 Categorical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Coniteness 216.1 Boolean algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.2 Conite topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6.2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.2.2 Double-pointed conite topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.3.1 Product topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.3.2 Direct sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7 Compactly generated space 237.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

8 Continuous function 268.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.2 Real-valued continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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8.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.2.3 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.2.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.2.5 Directional and semi-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

8.3 Continuous functions between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.3.1 Uniform, Hlder and Lipschitz continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 36

8.4 Continuous functions between topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.4.1 Alternative denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.4.3 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.4.4 Dening topologies via continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . 40

8.5 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9 Countable set 439.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.4 Formal denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.5 Minimal model of set theory is countable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499.6 Total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

10 Countably compact space 5110.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

11 Dense set 5211.1 Density in metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5211.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5211.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.4 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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11.6.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.6.2 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

12 First-countable space 5512.1 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5512.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5512.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5612.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

13 Limit of a function 5713.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5713.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5713.3 Functions of a single variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

13.3.1 Existence and one-sided limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5813.3.2 More general subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5913.3.3 Deleted versus non-deleted limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6013.3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

13.4 Functions on metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6113.5 Functions on topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6213.6 Limits involving innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

13.6.1 Limits at innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6213.6.2 Innite limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6313.6.3 Alternative notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6413.6.4 Limits at innity for rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

13.7 Functions of more than one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.8 Sequential limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.9 Other characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

13.9.1 In terms of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.9.2 In non-standard calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.9.3 In terms of nearness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

13.10Relationship to continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6713.11Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

13.11.1 Chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6813.11.2 Limits of special interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6813.11.3 L'Hpitals rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.11.4 Summations and integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

13.12See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7013.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

14 Limit of a sequence 7214.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7214.2 Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

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14.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7314.2.2 Formal Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7414.2.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7414.2.4 Innite limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

14.3 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.3.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

14.4 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.4.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

14.5 Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.6 Denition in hyperreal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7614.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

14.8.1 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7614.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

15 Lindelf space 7815.1 Properties of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7815.2 Properties of strongly Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7815.3 Product of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7815.4 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

16 Mathematical object 8016.1 Cantorian framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8016.2 Foundational paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8016.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8116.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8116.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8116.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

17 Mathematics 8217.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

17.1.1 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8317.1.2 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

17.2 Denitions of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8617.3 Inspiration, pure and applied mathematics, and aesthetics . . . . . . . . . . . . . . . . . . . . . . . 8817.4 Notation, language, and rigor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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17.5 Fields of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8917.5.1 Foundations and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8917.5.2 Pure mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9117.5.3 Applied mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

17.6 Mathematical awards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9317.7 Mathematics as science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9317.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9517.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9517.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9717.11Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9817.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

18 Neighbourhood (mathematics) 10018.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10118.2 In a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10118.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10318.4 Topology from neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10318.5 Uniform neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10318.6 Deleted neighbourhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10318.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10318.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

19 Neighbourhood system 10519.1 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10519.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10519.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10519.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10619.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

20 Point (geometry) 10720.1 Points in Euclidean geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10720.2 Dimension of a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

20.2.1 Vector space dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10820.2.2 Topological dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10820.2.3 Hausdor dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

20.3 Geometry without points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10920.4 Point masses and the Dirac delta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10920.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10920.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11020.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

21 Quotient space (topology) 11121.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

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21.2 Quotient map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11221.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11221.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11321.5 Compatibility with other topological notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11421.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

21.6.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11421.6.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

21.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

22 Second-countable space 11522.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

22.1.1 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11522.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11622.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

23 Separable space 11723.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11723.2 Separability versus second countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11723.3 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11823.4 Constructive mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11823.5 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

23.5.1 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11823.5.2 Non-separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

23.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11923.6.1 Embedding separable metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

23.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

24 Sequence 12124.1 Examples and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

24.1.1 Important examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12224.1.2 Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12324.1.3 Specifying a sequence by recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

24.2 Formal denition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12424.2.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12424.2.2 Finite and innite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12524.2.3 Increasing and decreasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12524.2.4 Bounded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12524.2.5 Other types of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

24.3 Limits and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12624.3.1 Denition of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12724.3.2 Applications and important results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12724.3.3 Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

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24.4 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12824.5 Use in other elds of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

24.5.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12924.5.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12924.5.3 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13024.5.4 Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13024.5.5 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13124.5.6 Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13124.5.7 Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

24.6 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13124.7 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13224.8 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13224.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13224.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13224.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

25 Sequential space 13425.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13425.2 Sequential closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13425.3 FrchetUrysohn space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13525.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13525.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13525.6 Equivalent conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13625.7 Categorical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13625.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13625.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

26 Sequentially compact space 13826.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13826.2 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13826.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13826.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13826.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

27 Subset 14027.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14127.2 and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14127.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14127.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14227.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14227.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14227.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

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28 Topological space 14428.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

28.1.1 Neighbourhoods denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14428.1.2 Open sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14528.1.3 Closed sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14628.1.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

28.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14628.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14628.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14728.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14828.6 Classication of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14828.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14828.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14828.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14828.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14928.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14928.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14928.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

29 Topology 15129.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15229.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15329.3 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

29.3.1 Topologies on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15529.3.2 Continuous functions and homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 15629.3.3 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

29.4 Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15629.4.1 General topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15629.4.2 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15729.4.3 Dierential topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15729.4.4 Geometric topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15729.4.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

29.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15829.5.1 Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15829.5.2 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15829.5.3 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15829.5.4 Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

29.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15829.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15929.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16029.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

x CONTENTS

30 Without loss of generality 16130.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16130.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16130.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16130.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

31 -compact space 16331.1 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16331.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16331.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16431.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16431.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 165

31.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16531.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17131.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Chapter 1

Axiom of countability

In mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) thatasserts the existence of a countable set with certain properties. Without such an axiom, such a set might not exist.

1.1 Important examplesImportant countability axioms for topological spaces include:[1]

sequential space: a set is open if every sequence convergent to a point in the set is eventually in the set

rst-countable space: every point has a countable neighbourhood basis (local base)

second-countable space: the topology has a countable base

separable space: there exists a countable dense subspace

Lindelf space: every open cover has a countable subcover

-compact space: there exists a countable cover by compact spaces

1.2 Relationships with each otherThese axioms are related to each other in the following ways:

Every rst countable space is sequential.

Every second-countable space is rst-countable, separable, and Lindelf.

Every -compact space is Lindelf.

Every metric space is rst countable.

For metric spaces second-countability, separability, and the Lindelf property are all equivalent.

1.3 Related conceptsOther examples of mathematical objects obeying axioms of innity include sigma-nite measure spaces, and latticesof countable type.

1

2 CHAPTER 1. AXIOM OF COUNTABILITY

1.4 References[1] Nagata, J.-I. (1985), Modern General Topology, North-Holland Mathematical Library (3rd ed.), Elsevier, p. 104, ISBN

9780080933795.

Chapter 2

Ball (mathematics)

N-ball redirects here. For the video game, see N-ball (game).In mathematics, a ball is the space inside a sphere. It may be a closed ball (including the boundary points of the

In Euclidean space, a ball is the inside of a sphere

sphere) or an open ball (excluding them).

3

4 CHAPTER 2. BALL (MATHEMATICS)

These concepts are dened not only in three-dimensional Euclidean space but also for lower and higher dimensions,and for metric spaces in general. A ball in n dimensions is called an n-ball and is bounded by an (n-1)-sphere. Thus,for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean3-space, a ball is taken to be the volume bounded by a 2-dimensional spherical shell boundary.In other contexts, such as in Euclidean geometry and informal use, sphere is sometimes used to mean ball.

2.1 Balls in Euclidean spaceIn Euclidean n-space, an (open) n-ball of radius r and center x is the set of all points of distance < r from x. A closedn-ball of radius r is the set of all points of distance r away from x.In Euclidean n-space, every ball is the interior of a hypersphere (a hyperball), that is a bounded interval when n = 1,the interior of a circle (a disk) when n = 2, and the interior of a sphere when n = 3.

2.1.1 The volumeMain article: Volume of an n-ball

The n-dimensional volume of a Euclidean ball of radius R in n-dimensional Euclidean space is:[1]

Vn(R) =n/2

(n2 + 1)Rn;

where is Leonhard Euler's gamma function (which can be thought of as an extension of the factorial function tofractional arguments). Using explicit formulas for particular values of the gamma function at the integers and halfintegers gives formulas for the volume of a Euclidean ball that do not require an evaluation of the gamma function.These are:

V2k(R) =k

k!R2k;

V2k+1(R) =2k+1k

(2k + 1)!!R2k+1 =

2(k!)(4)k

(2k + 1)!R2k+1:

In the formula for odd-dimensional volumes, the double factorial (2k + 1)!! is dened for odd integers 2k + 1 as (2k+ 1)!! = 1 3 5 (2k 1) (2k + 1).

2.2 Balls in general metric spacesLet (M,d) be a metric space, namely a set M with a metric (distance function) d. The open (metric) ball of radius r> 0 centered at a point p in M, usually denoted by B(p) or B(p; r), is dened by

Br(p) , fx 2M j d(x; p) < rg:

The closed (metric) ball, which may be denoted by B[p] or B[p; r], is dened by

Br[p] , fx 2M j d(x; p) rg:

Note in particular that a ball (open or closed) always includes p itself, since the denition requires r > 0.

2.3. BALLS IN NORMED VECTOR SPACES 5

The closure of the open ball B(p) is usually denoted Br(p) . While it is always the case that Br(p) Br(p) andBr(p) Br[p] , it is not always the case that Br(p) = Br[p] . For example, in a metric space X with the discretemetric, one has B1(p) = fpg and B1[p] = X , for any p 2 X .A (open or closed) unit ball is a ball of radius 1.A subset of a metric space is bounded if it is contained in some ball. A set is totally bounded if, given any positiveradius, it is covered by nitely many balls of that radius.The open balls of a metric space are a basis for a topological space, whose open sets are all possible unions of openballs. This space is called the topology induced by the metric d.

2.3 Balls in normed vector spacesAny normed vector space V with norm || is also a metric space, with the metric d(x, y) = |x y|. In such spaces,every ball B(p) is a copy of the unit ball B1(0), scaled by r and translated by p.The Euclidean balls discussed earlier are an example of balls in a normed vector space.

2.3.1 p-normIn Cartesian space Rn with the p-norm L, an open ball, is the set

B(r) =

(x 2 Rn :

nXi=1

jxijp < rp):

For n=2, in particular, the balls of L1 (often called the taxicab or Manhattan metric) are squares with the diagonalsparallel to the coordinate axes; those of L (the Chebyshev metric) are squares with the sides parallel to the coordinateaxes. For other values of p, the balls are the interiors of Lam curves (hypoellipses or hyperellipses).For n = 3, the balls of L1 are octahedra with axis-aligned body diagonals, those of L are cubes with axis-alignededges, and those of L with p > 2 are superellipsoids.

2.3.2 General convex normMore generally, given any centrally symmetric, bounded, open, and convex subset X of Rn, one can dene a normon Rn where the balls are all translated and uniformly scaled copies of X. Note this theorem does not hold if opensubset is replaced by closed subset, because the origin point qualies but does not dene a norm on Rn.

2.4 Topological ballsOne may talk about balls in any topological space X, not necessarily induced by a metric. An (open or closed) n-dimensional topological ball of X is any subset of X which is homeomorphic to an (open or closed) Euclidean n-ball.Topological n-balls are important in combinatorial topology, as the building blocks of cell complexes.Any open topological n-ball is homeomorphic to the Cartesian space Rn and to the open unit n-cube (hypercube)(0; 1)n Rn . Any closed topological n-ball is homeomorphic to the closed n-cube [0, 1]n.An n-ball is homeomorphic to an m-ball if and only if n = m. The homeomorphisms between an open n-ball B andRn can be classied in two classes, that can be identied with the two possible topological orientations of B.A topological n-ball need not be smooth; if it is smooth, it need not be dieomorphic to a Euclidean n-ball.

2.5 See also Ball - ordinary meaning

6 CHAPTER 2. BALL (MATHEMATICS)

Disk (mathematics) Formal ball, an extension to negative radii Neighborhood (mathematics) 3-sphere n-sphere, or hypersphere Alexander horned sphere Manifold Volume of an n-ball

2.6 References[1] Equation 5.19.4, NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.6 of 2013-05-06.

D. J. Smith and M. K. Vamanamurthy, How small is a unit ball?", Mathematics Magazine, 62 (1989) 101107. Robin conditions on the Euclidean ball, J. S. Dowker Isometries of the space of convex bodies contained in a Euclidean ball, Peter M. Gruber

Chapter 3

Base (topology)

In mathematics, a base (or basis) B for a topological spaceX with topology T is a collection of open sets in T such thatevery open set in T can be written as a union of elements of B.[1][2][note 1] We say that the base generates the topologyT. Bases are useful because many properties of topologies can be reduced to statements about a base generating thattopology, and because many topologies are most easily dened in terms of a base which generates them.

3.1 Simple properties of bases

Two important properties of bases are:

1. The base elements cover X.

2. Let B1, B2 be base elements and let I be their intersection. Then for each x in I, there is a base element B3containing x and contained in I.

If a collection B of subsets of X fails to satisfy either of these, then it is not a base for any topology on X. (It is asubbase, however, as is any collection of subsets of X.) Conversely, if B satises both of the conditions 1 and 2, thenthere is a unique topology on X for which B is a base; it is called the topology generated by B. (This topology is theintersection of all topologies on X containing B.) This is a very common way of dening topologies. A sucient butnot necessary condition for B to generate a topology on X is that B is closed under intersections; then we can alwaystake B3 = I above.For example, the collection of all open intervals in the real line forms a base for a topology on the real line becausethe intersection of any two open intervals is itself an open interval or empty. In fact they are a base for the standardtopology on the real numbers.However, a base is not unique. Many bases, even of dierent sizes, may generate the same topology. For example,the open intervals with rational endpoints are also a base for the standard real topology, as are the open intervals withirrational endpoints, but these two sets are completely disjoint and both properly contained in the base of all openintervals. In contrast to a basis of a vector space in linear algebra, a base need not be maximal; indeed, the onlymaximal base is the topology itself. In fact, any open set generated by a base may be safely added to the base withoutchanging the topology. The smallest possible cardinality of a base is called the weight of the topological space.An example of a collection of open sets which is not a base is the set S of all semi-innite intervals of the forms (,a) and (a, ), where a is a real number. Then S is not a base for any topology on R. To show this, suppose it were.Then, for example, (, 1) and (0, ) would be in the topology generated by S, being unions of a single base element,and so their intersection (0,1) would be as well. But (0, 1) clearly cannot be written as a union of the elements of S.Using the alternate denition, the second property fails, since no base element can t inside this intersection.Given a base for a topology, in order to prove convergence of a net or sequence it is sucient to prove that it iseventually in every set in the base which contains the putative limit.

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8 CHAPTER 3. BASE (TOPOLOGY)

3.2 Objects dened in terms of bases The order topology is usually dened as the topology generated by a collection of open-interval-like sets. The metric topology is usually dened as the topology generated by a collection of open balls. A second-countable space is one that has a countable base. The discrete topology has the singletons as a base. The pronite topology on a group is dened by taking the collection of all normal subgroups of nite index as

a basis of open neighborhoods of the identity.

3.3 Theorems For each point x in an open set U, there is a base element containing x and contained in U. A topology T2 is ner than a topology T1 if and only if for each x and each base element B of T1 containingx, there is a base element of T2 containing x and contained in B.

If B1,B2,...,Bn are bases for the topologies T1,T2,...,Tn, then the set product B1 B2 ... Bn is a base forthe product topology T1 T2 ... Tn. In the case of an innite product, this still applies, except that all butnitely many of the base elements must be the entire space.

Let B be a base for X and let Y be a subspace of X. Then if we intersect each element of B with Y, the resultingcollection of sets is a base for the subspace Y.

If a function f:X Y maps every base element of X into an open set of Y, it is an open map. Similarly, ifevery preimage of a base element of Y is open in X, then f is continuous.

A collection of subsets of X is a topology on X if and only if it generates itself. B is a basis for a topological space X if and only if the subcollection of elements of B which contain x form a

local base at x, for any point x of X.

3.4 Base for the closed setsClosed sets are equally adept at describing the topology of a space. There is, therefore, a dual notion of a base for theclosed sets of a topological space. Given a topological space X, a family of closed sets F forms a base for the closedsets if and only if for each closed set A and each point x not in A there exists an element of F containing A but notcontaining x.It is easy to check that F is a base for the closed sets of X if and only if the family of complements of members of Fis a base for the open sets of X.Let F be a base for the closed sets of X. Then

1. F =

2. For each F1 and F2 in F the union F1 F2 is the intersection of some subfamily of F (i.e. for any x not in F1or F2 there is an F3 in F containing F1 F2 and not containing x).

Any collection of subsets of a set X satisfying these properties forms a base for the closed sets of a topology on X.The closed sets of this topology are precisely the intersections of members of F.In some cases it is more convenient to use a base for the closed sets rather than the open ones. For example, a spaceis completely regular if and only if the zero sets form a base for the closed sets. Given any topological space X, thezero sets form the base for the closed sets of some topology on X. This topology will be the nest completely regulartopology on X coarser than the original one. In a similar vein, the Zariski topology on An is dened by taking thezero sets of polynomial functions as a base for the closed sets.

3.5. WEIGHT AND CHARACTER 9

3.5 Weight and characterWe shall work with notions established in (Engelking 1977, p. 12, pp. 127-128).Fix X a topological space. We dene the weight, w(X), as the minimum cardinality of a basis; we dene the networkweight, nw(X), as the minimum cardinality of a network; the character of a point, (x;X) , as the minimumcardinality of a neighbourhood basis for x in X; and the character of X to be

(X) , supf(x;X) : x 2 Xg:Here, a network is a family N of sets, for which, for all points x and open neighbourhoods U containing x, thereexists B in N for which x B U.The point of computing the character and weight is useful to be able to tell what sort of bases and local bases canexist. We have following facts:

nw(X) w(X). if X is discrete, then w(X) = nw(X) = |X|. if X is Hausdor, then nw(X) is nite i X is nite discrete. if B a basis of X then there is a basis B0 B of size jB0j w(X) . if N a neighbourhood basis for x in X then there is a neighbourhood basis N 0 N of size jN 0j (x;X) . if f : X Y is a continuous surjection, then nw(Y) w(X). (Simply consider the Y-network f 000B , ff 00U :U 2 Bg for each basis B of X.)

if (X; ) is Hausdor, then there exists a weaker Hausdor topology (X; 0) so that w(X; 0) nw(X; ) .So a fortiori, if X is also compact, then such topologies coincide and hence we have, combined with the rstfact, nw(X) = w(X).

if f : X Y a continuous surjective map from a compact metrisable space to an Hausdor space, then Y iscompact metrisable.

The last fact follows from f(X) being compact Hausdor, and hence nw(f(X)) = w(f(X)) w(X) @0 (sincecompact metrisable spaces are necessarily second countable); as well as the fact that compact Hausdor spaces aremetrisable exactly in case they are second countable. (An application of this, for instance, is that every path in anHausdor space is compact metrisable.)

3.5.1 Increasing chains of open setsUsing the above notation, suppose that w(X) some innite cardinal. Then there does not exist a strictly increasingsequence of open sets (equivalently strictly decreasing sequence of closed sets) of length +.To see this (without the axiom of choice), x

fUg2 ;as a basis of open sets. And suppose per contra, that

fVg2+were a strictly increasing sequence of open sets. This means

8 < + : V n[

10 CHAPTER 3. BASE (TOPOLOGY)

For

x 2 V n[

Chapter 4

Category (mathematics)

This is a category with a collection of objects A, B, C and collection of morphisms denoted f, g, g f, and the loops are the identityarrows. This category is typically denoted by a boldface 3.

In mathematics, a category is an algebraic structure that comprises objects that are linked by arrows. A categoryhas two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for

11

12 CHAPTER 4. CATEGORY (MATHEMATICS)

each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. Onthe other hand, any monoid can be understood as a special sort of category, and so can any preorder. In general,the objects and arrows may be abstract entities of any kind, and the notion of category provides a fundamental andabstract way to describe mathematical entities and their relationships. This is the central idea of category theory, abranch of mathematics which seeks to generalize all of mathematics in terms of objects and arrows, independent ofwhat the objects and arrows represent. Virtually every branch of modern mathematics can be described in terms ofcategories, and doing so often reveals deep insights and similarities between seemingly dierent areas of mathematics.For more extensive motivational background and historical notes, see category theory and the list of category theorytopics.Two categories are the same if they have the same collection of objects, the same collection of arrows, and thesame associative method of composing any pair of arrows. Two categories may also be considered "equivalent" forpurposes of category theory, even if they are not precisely the same.Well-known categories are denoted by a short capitalized word or abbreviation in bold or italics: examples include Set,the category of sets and set functions; Ring, the category of rings and ring homomorphisms; and Top, the categoryof topological spaces and continuous maps. All of the preceding categories have the identity map as identity arrowand composition as the associative operation on arrows.The classic and still much used text on category theory is Categories for the Working Mathematician by Saunders MacLane. Other references are given in the References below. The basic denitions in this article are contained withinthe rst few chapters of any of these books.

4.1 DenitionThere are many equivalent denitions of a category.[1] One commonly used denition is as follows. A category Cconsists of

a class ob(C) of objects

a class hom(C) ofmorphisms, or arrows, ormaps, between the objects. Each morphism f has a unique sourceobject a and target object b where a and b are in ob(C). We write f: a b, and we say "f is a morphism froma to b". We write hom(a, b) (or homC(a, b) when there may be confusion about to which category hom(a, b)refers) to denote the hom-class of all morphisms from a to b. (Some authors write Mor(a, b) or simply C(a,b) instead.)

for every three objects a, b and c, a binary operation hom(a, b) hom(b, c) hom(a, c) called compositionof morphisms; the composition of f : a b and g : b c is written as g f or gf. (Some authors usediagrammatic order, writing f;g or fg.)

such that the following axioms hold:

(associativity) if f : a b, g : b c and h : c d then h (g f) = (h g) f, and

(identity) for every object x, there exists a morphism 1x : x x (some authors write idx) called the identitymorphism for x, such that for every morphism f : a x and every morphism g : x b, we have 1x f = f andg 1x = g.

From these axioms, one can prove that there is exactly one identity morphism for every object. Some authors use aslight variation of the denition in which each object is identied with the corresponding identity morphism.

4.2 HistoryCategory theory rst appeared in a paper entitled General Theory of Natural Equivalences, written by SamuelEilenberg and Saunders Mac Lane in 1945.

4.3. SMALL AND LARGE CATEGORIES 13

4.3 Small and large categoriesA category C is called small if both ob(C) and hom(C) are actually sets and not proper classes, and large otherwise.A locally small category is a category such that for all objects a and b, the hom-class hom(a, b) is a set, called ahomset. Many important categories in mathematics (such as the category of sets), although not small, are at leastlocally small.

4.4 ExamplesThe class of all sets together with all functions between sets, where composition is the usual function composition,forms a large category, Set. It is the most basic and the most commonly used category in mathematics. The categoryRel consists of all sets, with binary relations as morphisms. Abstracting from relations instead of functions yieldsallegories, a special class of categories.Any class can be viewed as a category whose only morphisms are the identity morphisms. Such categories are calleddiscrete. For any given set I, the discrete category on I is the small category that has the elements of I as objects andonly the identity morphisms as morphisms. Discrete categories are the simplest kind of category.Any preordered set (P, ) forms a small category, where the objects are the members of P, the morphisms are arrowspointing from x to y when x y. Between any two objects there can be at most one morphism. The existence ofidentity morphisms and the composability of the morphisms are guaranteed by the reexivity and the transitivity ofthe preorder. By the same argument, any partially ordered set and any equivalence relation can be seen as a smallcategory. Any ordinal number can be seen as a category when viewed as an ordered set.Any monoid (any algebraic structure with a single associative binary operation and an identity element) forms a smallcategory with a single object x. (Here, x is any xed set.) The morphisms from x to x are precisely the elements ofthe monoid, the identity morphism of x is the identity of the monoid, and the categorical composition of morphismsis given by the monoid operation. Several denitions and theorems about monoids may be generalized for categories.Any group can be seen as a category with a single object in which every morphism is invertible (for every morphismf there is a morphism g that is both left and right inverse to f under composition) by viewing the group as acting onitself by left multiplication. A morphism which is invertible in this sense is called an isomorphism.A groupoid is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups, groupactions and equivalence relations.Any directed graph generates a small category: the objects are the vertices of the graph, and the morphisms are thepaths in the graph (augmented with loops as needed) where composition of morphisms is concatenation of paths.Such a category is called the free category generated by the graph.The class of all preordered sets with monotonic functions as morphisms forms a category, Ord. It is a concretecategory, i.e. a category obtained by adding some type of structure onto Set, and requiring that morphisms arefunctions that respect this added structure.The class of all groups with group homomorphisms as morphisms and function composition as the compositionoperation forms a large category, Grp. Like Ord, Grp is a concrete category. The category Ab, consisting ofall abelian groups and their group homomorphisms, is a full subcategory of Grp, and the prototype of an abeliancategory. Other examples of concrete categories are given by the following table.Fiber bundles with bundle maps between them form a concrete category.The category Cat consists of all small categories, with functors between them as morphisms.

4.5 Construction of new categories

4.5.1 Dual category

Any category C can itself be considered as a new category in a dierent way: the objects are the same as those inthe original category but the arrows are those of the original category reversed. This is called the dual or oppositecategory and is denoted Cop.

14 CHAPTER 4. CATEGORY (MATHEMATICS)

A directed graph.

4.5.2 Product categoriesIf C and D are categories, one can form the product category C D: the objects are pairs consisting of one objectfrom C and one from D, and the morphisms are also pairs, consisting of one morphism in C and one in D. Such pairscan be composed componentwise.

4.6 Types of morphismsA morphism f : a b is called

a monomorphism (or monic) if fg1 = fg2 implies g1 = g2 for all morphisms g1, g2 : x a. an epimorphism (or epic) if g1f = g2f implies g1 = g2 for all morphisms g1, g2 : b x. a bimorphism if it is both a monomorphism and an epimorphism. a retraction if it has a right inverse, i.e. if there exists a morphism g : b a with fg = 1b. a section if it has a left inverse, i.e. if there exists a morphism g : b a with gf = 1a. an isomorphism if it has an inverse, i.e. if there exists a morphism g : b a with fg = 1b and gf = 1a. an endomorphism if a = b. The class of endomorphisms of a is denoted end(a).

4.7. TYPES OF CATEGORIES 15

an automorphism if f is both an endomorphism and an isomorphism. The class of automorphisms of a isdenoted aut(a).

Every retraction is an epimorphism. Every section is a monomorphism. The following three statements are equivalent:

f is a monomorphism and a retraction; f is an epimorphism and a section; f is an isomorphism.

Relations among morphisms (such as fg = h) can most conveniently be represented with commutative diagrams, wherethe objects are represented as points and the morphisms as arrows.

4.7 Types of categories In many categories, e.g. Ab or VectK, the hom-sets hom(a, b) are not just sets but actually abelian groups,

and the composition of morphisms is compatible with these group structures; i.e. is bilinear. Such a cate-gory is called preadditive. If, furthermore, the category has all nite products and coproducts, it is called anadditive category. If all morphisms have a kernel and a cokernel, and all epimorphisms are cokernels and allmonomorphisms are kernels, then we speak of an abelian category. A typical example of an abelian categoryis the category of abelian groups.

A category is called complete if all limits exist in it. The categories of sets, abelian groups and topologicalspaces are complete.

A category is called cartesian closed if it has nite direct products and a morphism dened on a nite productcan always be represented by a morphism dened on just one of the factors. Examples include Set and CPO,the category of complete partial orders with Scott-continuous functions.

A topos is a certain type of cartesian closed category in which all of mathematics can be formulated (just likeclassically all of mathematics is formulated in the category of sets). A topos can also be used to represent alogical theory.

4.8 See also Enriched category Higher category theory Quantaloid Table of mathematical symbols

4.9 Notes[1] Barr & Wells, Chapter 1.

4.10 References Admek, Ji; Herrlich, Horst; Strecker, George E. (1990), Abstract and Concrete Categories (PDF), John Wiley

& Sons, ISBN 0-471-60922-6 (now free on-line edition, GNU FDL).

Asperti, Andrea; Longo, Giuseppe (1991), Categories, Types and Structures (PDF), MIT Press, ISBN 0-262-01125-5.

16 CHAPTER 4. CATEGORY (MATHEMATICS)

Awodey, Steve (2006), Category theory, Oxford logic guides 49, Oxford University Press, ISBN 978-0-19-856861-2.

Barr, Michael; Wells, Charles (2005), Toposes, Triples and Theories, Reprints in Theory and Applications ofCategories 12 (revised ed.), MR 2178101.

Borceux, Francis (1994), Handbook of Categorical Algebra, Encyclopedia of Mathematics and its Applica-tions, 5052, Cambridge: Cambridge University Press, ISBN 0-521-06119-9.

Hazewinkel, Michiel, ed. (2001), Category, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Herrlich, Horst; Strecker, George E. (2007), Category Theory, Heldermann Verlag. Jacobson, Nathan (2009), Basic algebra (2nd ed.), Dover, ISBN 978-0-486-47187-7. Lawvere, William; Schanuel, Steve (1997), Conceptual Mathematics: A First Introduction to Categories, Cam-

bridge: Cambridge University Press, ISBN 0-521-47249-0.

Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics 5 (2nded.), Springer-Verlag, ISBN 0-387-98403-8.

Marquis, Jean-Pierre (2006), Category Theory, in Zalta, Edward N., Stanford Encyclopedia of Philosophy. Sica, Giandomenico (2006), What is category theory?, Advanced studies in mathematics and logic 3, Polimet-

rica, ISBN 978-88-7699-031-1. category in nLab

Chapter 5

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.

5.1 Denitions

5.1.1 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.

5.1.2 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).

5.1.3 Closure of a setSee also: Closure (mathematics)

17

18 CHAPTER 5. CLOSURE (TOPOLOGY)

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.

5.2 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 say

that 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 is

equivalent 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.

5.3. CLOSURE OPERATOR 19

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 closure

of 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:

5.3 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.

5.4 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 the

closures 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 of

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

20 CHAPTER 5. CLOSURE (TOPOLOGY)

5.5 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.

5.6 See also Closure algebra

5.7 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

5.8 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

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

55608-010-4

Chapter 6

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.

6.1 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.

6.2 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.

6.2.1 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 and

sequentially 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

21

22 CHAPTER 6. COFINITENESS

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).

6.2.2 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.

6.3 Other examples

6.3.1 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.

6.3.2 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.

6.4 References Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of

1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446 (See example 18)

Chapter 7

Compactly generated space

In topology, a compactly generated space (or k-space) is a topological space whose topology is coherent with thefamily of all compact subspaces. Specically, a topological space X is compactly generated if it satises the followingcondition:

A subspace A is closed in X if and only if A K is closed in K for all compact subspaces K X.

Equivalently, one can replace closed with open in this denition. If X is coherent with any cover of compact subspacesin the above sense then it is, in fact, coherent with all compact subspaces.A compactly generated Hausdor space is a compactly generated space which is also Hausdor. Like manycompactness conditions, compactly generated spaces are often assumed to be Hausdor.

7.1 MotivationCompactly generated spaces were originally called k-spaces, after the German word kompakt. They were studied byHurewicz, and can be found in General Topology by Kelley, Topology by Dugundji, Rational Homotopy Theory byFlix, Halperin, Thomas.The motivation for their deeper study came in the 1960s from well known deciencies of the usual topological cate-gory. This fails to be a cartesian closed category, the usual cartesian product of identication maps is not always anidentication map, and the usual product of CW-complexes need not be a CW-complex. By contrast, the category ofsimplicial sets had many convenient properties, including being cartesian closed. The history of the study of repairingthis situation is given in the article on the ncatlab on convenient categories of spaces.The rst suggestion (1962) to remedy this situation was to restrict oneself to the full subcategory of compactly gen-erated Hausdor spaces, which is in fact Cartesian closed. These ideas extend on the de Vries duality theorem. Adenition of the exponential object is given below. Another suggestion (1964) was to consider the usual Hausdorspaces but use functions continuous on compact subsets.These ideas can be generalised to the non-Hausdor case, see section 5.9 in the book Topology and groupoids. Thisis useful since identication spaces of Hausdor spaces need not be Hausdor. For more information, see also thearticle by Booth and Tillotson.

7.2 ExamplesMost topological spaces commonly studied in mathematics are compactly generated.

Every compact space is compactly generated. Every locally compact space is compactly generated. Every rst-countable space is compactly generated.

23

24 CHAPTER 7. COMPACTLY GENERATED SPACE

Topological manifolds are locally compact Hausdor and therefore compactly generated Hausdor. Metric spaces are rst-countable and therefore compactly generated Hausdor. Every CW complex is compactly generated Hausdor.

7.3 PropertiesWe denote CGTop the full subcategory of Top with objects the compactly generated spaces, and CGHaus the fullsubcategory of CGTop with objects the Hausdor separated spaces.Given any topological space X we can dene a (possibly) ner topology on X which is compactly generated. Let{K} denote the family of compact subsets of X. We dene the new topology on X by declaring a subset A to beclosed if and only if A K is closed in K for each . Denote this new space by X. One can show that the compactsubsets of X and X coincide and the induced topologies are the same. It follows that X is compactly generated. IfX was compactly generated to start with then X = X otherwise the topology on X is strictly ner than X (i.e. thereare more open sets).This construction is functorial. The functor from Top to CGTop which takes X to X is right adjoint to the inclusionfunctor CGTop Top.The continuity of a map dened on compactly generated space X can be determined solely by looking at the compactsubsets of X. Specically, a function f : X Y is continuous if and only if it is continuous when restricted to eachcompact subset K X.If X and Y are two compactly generated spaces the product X Y may not be compactly generated (it will be if atleast one of the factors is locally compact). Therefore when working in categories of compactly generated spaces itis necessary to dene the product as (X Y).The exponential object in the CGHaus is given by (YX) where YX is the space of continuous maps from X to Y withthe compact-open topology.These ideas can be generalised to the non-Hausdor case, see section 5.9 in the book `Topology and groupoids listedbelow. This is useful since identication spaces of Hausdor spaces need not be Hausdor.

7.4 See also compact-open topology CW complex nitely generated space countably generated space Weak Hausdor space

7.5 References Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5

((2nd ed.) ed.). Springer-Verlag. ISBN 0-387-98403-8.

Willard, Stephen (1970). General Topology. Reading, Massachusetts: Addison-Wesley. ISBN 0-486-43479-6.

Brown, Ronald (2006). Topology and Groupoids. Charlottsville, N. Carolina: Booksurge. ISBN 1-4196-2722-8.

P. I. Booth and J. Tillotson, Monoidal Closed Categories and Convenient Categories of Topological Spaces,Pacic Journal of Mathematics, 88 (1980) 33-53.

7.5. REFERENCES 25

Strickland, Neil P. (2009). The category of CGWH spaces (PDF). Convenient category of topological spaces in nLab

Chapter 8

Continuous function

In mathematics, a continuous function is, roughly speaking, a function for which small changes in the input resultin small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous functionwith a continuous inverse function is called a homeomorphism.Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The intro-ductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers.In addition, this article discusses the denition for the more general case of functions between two metric spaces.In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Otherforms of continuity do exist but they are not discussed in this article.As an example, consider the function h(t), which describes the height of a growing ower at time t. This function iscontinuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumpswhenever money is deposited or withdrawn, so the function M(t) is discontinuous.

8.1 HistoryA form of this epsilon-delta denition of continuity was rst given by Bernard Bolzano in 1817. Augustin-LouisCauchy dened continuity of y = f(x) as follows: an innitely small increment of the independent variable xalways produces an innitely small change f(x+ ) f(x) of the dependent variable y (see e.g., Cours d'Analyse,p. 34). Cauchy dened innitely small quantities in terms of variable quantities, and his denition of continuityclosely parallels the innitesimal denition used today (see microcontinuity). The formal denition and the distinctionbetween pointwise continuity and uniform continuity were rst given by Bolzano in the 1830s but the work wasn'tpublished until the 1930s. Eduard Heine provided the rst published denition of uniform continuity in 1872, butbased these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854.[1]

8.2 Real-valued continuous functions

8.2.1 DenitionA function from the set of real numbers to the real numbers can be represented by a graph in the Cartesian plane;such a function is continuous if, roughly speaking, the graph is a single unbroken curve with no holes or jumps.There are several ways to make this denition mathematically rigorous. These denitions are equivalent to one an-other, so the most convenient denition can be used to determine whether a given function is continuous or not. Inthe denitions below,

f : I ! R:

is a function dened on a subset I of the set R of real numbers. This subset I is referred to as the domain of f. Somepossible choices include I=R, the whole set of real numbers, an open interval

26

8.2. REAL-VALUED CONTINUOUS FUNCTIONS 27

I = (a; b) = fx 2 R j a < x < bg;or a closed interval

I = [a; b] = fx 2 R j a x bg:Here, a and b are real numbers.

Denition in terms of limits of functions

The function f is continuous at some point c of its domain if the limit of f(x) as x approaches c through the domainof f exists and is equal to f(c).[2] In mathematical notation, this is written as

limx!c f(x) = f