Banach spaceFrom Wikipedia, the free encyclopedia

Contents

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

2 Baire space (set theory) 32.1 Topology and trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Relation to the real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Banach space 53.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2.1 Linear operators, isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.2 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.3 Classical spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.4 Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.5 Dual space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.6 Banachs theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.7 Reexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2.8 Weak convergences of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Schauder bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 Tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4.1 Tensor products and the approximation property . . . . . . . . . . . . . . . . . . . . . . . 153.5 Some classication results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.5.1 Characterizations of Hilbert space among Banach spaces . . . . . . . . . . . . . . . . . . 153.5.2 Spaces of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.7 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.8 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

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3.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Bijection 214.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2.1 Batting line-up of a baseball team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2.2 Seats and students of a classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 More mathematical examples and some non-examples . . . . . . . . . . . . . . . . . . . . . . . . 234.4 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.6 Bijections and cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.8 Bijections and category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.9 Generalization to partial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.10 Contrast with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Borel isomorphism 275.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Borel measure 286.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2 On the real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.3.1 LebesgueStieltjes integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3.2 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.3.3 Hausdor dimension and Frostmans lemma . . . . . . . . . . . . . . . . . . . . . . . . . 296.3.4 CramrWold theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7 Cantor space 317.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.2 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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7.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

8 Cardinal number 338.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.3 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.4 Cardinal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

8.4.1 Successor cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.4.2 Cardinal addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.4.3 Cardinal multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.4.4 Cardinal exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

8.5 The continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

9 Cardinality of the continuum 419.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9.1.1 Uncountability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.1.2 Cardinal equalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.1.3 Alternative explanation for c = 2@0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9.2 Beth numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.3 The continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.4 Sets with cardinality of the continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.5 Sets with greater cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10 Cauchy sequence 4610.1 In real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.2 In a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

10.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.3.2 Counter-example: rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.3.3 Counter-example: open interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.3.4 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

10.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.4.1 In topological vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.4.2 In topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.4.3 In groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.4.4 In constructive mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.4.5 In a hyperreal continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

10.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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10.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

11 Closure (topology) 5111.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

11.1.1 Point of closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.1.2 Limit point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.1.3 Closure of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5211.3 Closure operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.4 Facts about closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.5 Categorical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

12 Complete metric space 5512.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5512.2 Some theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5612.3 Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5612.4 Topologically complete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5712.5 Alternatives and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5712.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5712.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

13 Completely metrizable space 5913.1 Dierence between complete metric space and completely metrizable space . . . . . . . . . . . . . . 5913.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5913.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5913.4 Completely metrizable abelian topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . 6013.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6013.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6013.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

14 Continuous function 6214.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6214.2 Real-valued continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

14.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6214.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6514.2.3 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

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14.2.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6914.2.5 Directional and semi-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

14.3 Continuous functions between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7114.3.1 Uniform, Hlder and Lipschitz continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 72

14.4 Continuous functions between topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 7214.4.1 Alternative denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7414.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.4.3 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7614.4.4 Dening topologies via continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . 76

14.5 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7614.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

15 Countable set 7915.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.4 Formal denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8015.5 Minimal model of set theory is countable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8515.6 Total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8515.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8615.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8615.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8615.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

16 Countably compact space 8716.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8716.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8716.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8716.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

17 Cover (topology) 8817.1 Cover in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8817.2 Renement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8817.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8917.4 Covering dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8917.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8917.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9017.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9017.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

18 Dense set 91

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18.1 Density in metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9118.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9118.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9218.4 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9218.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9218.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

18.6.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9318.6.2 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

19 Descriptive set theory 9419.1 Polish spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

19.1.1 Universality properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9419.2 Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

19.2.1 Borel hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9519.2.2 Regularity properties of Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

19.3 Analytic and coanalytic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9519.4 Projective sets and Wadge degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9519.5 Borel equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9619.6 Eective descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9619.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9619.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9619.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

20 Discrete space 9720.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9720.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9720.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9920.4 Indiscrete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9920.5 Quotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9920.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9920.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

21 Disjoint sets 10021.1 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10021.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10121.3 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10121.4 Disjoint unions and partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10221.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10221.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10221.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

22 Ernst Leonard Lindelf 10422.1 Selected bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

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22.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10422.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

23 Euclidean space 10623.1 Intuitive overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10623.2 Euclidean structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

23.2.1 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10823.2.2 Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10923.2.3 Rotations and reections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10923.2.4 Euclidean group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

23.3 Non-Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11123.4 Geometric shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

23.4.1 Lines, planes, and other subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11223.4.2 Line segments and triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11323.4.3 Polytopes and root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11423.4.4 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11423.4.5 Balls, spheres, and hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

23.5 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11523.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11523.7 Alternatives and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

23.7.1 Curved spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11523.7.2 Indenite quadratic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11523.7.3 Other number elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11623.7.4 Innite dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

23.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11623.9 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11623.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11623.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

24 Exhaustion by compact sets 11724.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11724.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11724.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

25 Filter (mathematics) 11825.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11925.2 General denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11925.3 Filter on a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

25.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12025.3.2 Filters in model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12025.3.3 Filters in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

25.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

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25.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12325.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

26 Hausdor space 12426.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12426.2 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12526.3 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12526.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12526.5 Preregularity versus regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12626.6 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12626.7 Algebra of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12726.8 Academic humour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12726.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12726.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12726.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

27 Homeomorphism 12827.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12827.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

27.2.1 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12927.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13027.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13027.5 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13127.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13127.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13127.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

28 Inner regular measure 13228.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13228.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13228.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13228.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

29 Interval (mathematics) 13429.1 Notations for intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

29.1.1 Including or excluding endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13429.1.2 Innite endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13529.1.3 Integer intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

29.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13529.3 Classication of intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

29.3.1 Intervals of the extended real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13629.4 Properties of intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13629.5 Dyadic intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

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29.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13729.6.1 Multi-dimensional intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13729.6.2 Complex intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

29.7 Topological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13729.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13829.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13829.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

30 Johann Radon 13930.1 Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13930.2 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13930.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14030.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

31 Lawson topology 14131.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14131.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14131.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14131.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

32 Lexicographic order topology on the unit square 14232.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14232.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14232.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14232.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

33 Limit (mathematics) 14333.1 Limit of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14333.2 Limit of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14433.3 Limit as standard part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14533.4 Convergence and xed point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14533.5 Topological net . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14533.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14633.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14633.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

34 Limit of a sequence 14734.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14734.2 Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

34.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14834.2.2 Formal Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14934.2.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14934.2.4 Innite limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

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34.3 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15034.3.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15034.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

34.4 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15034.4.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15034.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

34.5 Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15034.6 Denition in hyperreal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15034.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15134.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

34.8.1 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15134.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15234.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

35 Limit point 15335.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15335.2 Types of limit points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15335.3 Some facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15435.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15435.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

36 Lindelf space 15636.1 Properties of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15636.2 Properties of strongly Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15636.3 Product of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15636.4 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15736.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15736.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15736.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

37 Lindelfs lemma 15837.1 Statement of the lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15837.2 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15837.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

38 List of examples in general topology 15938.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

39 Local homeomorphism 16139.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16139.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16139.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16139.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

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39.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

40 Local property 16340.1 Properties of a single space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

40.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16340.2 Properties of a pair of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16340.3 Properties of innite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16440.4 Properties of nite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16440.5 Properties of commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

41 Locally compact space 16541.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16541.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

41.2.1 Compact Hausdor spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16641.2.2 Locally compact Hausdor spaces that are not compact . . . . . . . . . . . . . . . . . . . 16641.2.3 Hausdor spaces that are not locally compact . . . . . . . . . . . . . . . . . . . . . . . . 16641.2.4 Non-Hausdor examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

41.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16741.3.1 The point at innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16741.3.2 Locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

41.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16841.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

42 Locally connected space 16942.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17042.2 Denitions and rst examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

42.2.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17142.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17142.4 Components and path components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

42.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17242.5 Quasicomponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

42.5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17242.6 More on local connectedness versus weak local connectedness . . . . . . . . . . . . . . . . . . . . 17342.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17342.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17342.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17442.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

43 Locally nite collection 17543.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

43.1.1 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17543.1.2 Second countable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

43.2 Closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

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43.3 Countably locally nite collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17643.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

44 Locally nite space 17744.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

45 Metric space 17845.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17845.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17845.3 Examples of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17945.4 Open and closed sets, topology and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 18045.5 Types of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

45.5.1 Complete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18045.5.2 Bounded and totally bounded spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18145.5.3 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18245.5.4 Locally compact and proper spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18245.5.5 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18245.5.6 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

45.6 Types of maps between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18245.6.1 Continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18345.6.2 Uniformly continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18345.6.3 Lipschitz-continuous maps and contractions . . . . . . . . . . . . . . . . . . . . . . . . . 18345.6.4 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18445.6.5 Quasi-isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

45.7 Notions of metric space equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18445.8 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18445.9 Distance between points and sets; Hausdor distance and Gromov metric . . . . . . . . . . . . . . 18545.10Product metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

45.10.1 Continuity of distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18545.11Quotient metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18645.12Generalizations of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

45.12.1 Metric spaces as enriched categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18645.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18745.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18745.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18845.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

46 Metrization theorem 18946.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18946.2 Metrization theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18946.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19046.4 Examples of non-metrizable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

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46.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19046.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

47 Neighbourhood (mathematics) 19147.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19247.2 In a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19247.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19447.4 Topology from neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19447.5 Uniform neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19447.6 Deleted neighbourhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19447.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19447.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

48 Neighbourhood system 19648.1 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19648.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19648.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19648.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19748.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

49 Net (mathematics) 19849.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19849.2 Examples of nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19849.3 Limits of nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19949.4 Examples of limits of nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19949.5 Supplementary denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19949.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19949.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20049.8 Cauchy nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20149.9 Relation to lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20149.10Limit superior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20149.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

50 Normed vector space 20350.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20350.2 Topological structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20450.3 Linear maps and dual spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20450.4 Normed spaces as quotient spaces of seminormed spaces . . . . . . . . . . . . . . . . . . . . . . . 20550.5 Finite product spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20550.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20650.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

51 Paracompact space 207

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51.1 Paracompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20751.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20751.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20851.4 Paracompact Hausdor Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

51.4.1 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20951.5 Relationship with compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

51.5.1 Comparison of properties with compactness . . . . . . . . . . . . . . . . . . . . . . . . . 21051.6 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

51.6.1 Denition of relevant terms for the variations . . . . . . . . . . . . . . . . . . . . . . . . . 21151.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21151.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21151.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21251.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

52 Polish space 21352.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21352.2 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21452.3 Polish metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21452.4 Generalizations of Polish spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

52.4.1 Lusin spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21452.4.2 Suslin spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21452.4.3 Radon spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21552.4.4 Polish groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

52.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21552.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

53 Probability measure 21653.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21753.2 Example applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21753.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21853.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21853.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

54 Probability theory 21954.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21954.2 Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

54.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21954.2.2 Discrete probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22054.2.3 Continuous probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22154.2.4 Measure-theoretic probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

54.3 Classical probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22254.4 Convergence of random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

CONTENTS xv

54.4.1 Law of large numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22354.4.2 Central limit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

54.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22454.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22454.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22554.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

55 Quotient space (topology) 22655.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22755.2 Quotient map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22755.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22755.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22855.5 Compatibility with other topological notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22955.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

55.6.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22955.6.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

55.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

56 Radon measure 23056.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23056.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23056.3 Radon measures on locally compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

56.3.1 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23156.3.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

56.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23256.5 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

56.5.1 Moderated Radon measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23356.5.2 Radon spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23356.5.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23356.5.4 Metric space structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

56.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23456.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

57 Radon space 23557.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

58 Second-countable space 23658.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

58.1.1 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23658.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23758.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

59 Separable space 238

xvi CONTENTS

59.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23859.2 Separability versus second countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23859.3 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23959.4 Constructive mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23959.5 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

59.5.1 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23959.5.2 Non-separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

59.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24059.6.1 Embedding separable metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

59.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

60 Separation axiom 24260.1 Preliminary denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24260.2 Main denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24360.3 Relationships between the axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24460.4 Other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24460.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24560.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24560.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

61 Subspace topology 24861.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24861.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24861.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24961.4 Preservation of topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25061.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25061.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

62 Topological space 25162.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

62.1.1 Neighbourhoods denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25162.1.2 Open sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25262.1.3 Closed sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25362.1.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

62.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25362.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25362.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25462.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25562.6 Classication of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25562.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25562.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25562.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

CONTENTS xvii

62.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25662.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25662.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25662.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

63 Uncountable set 25863.1 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25863.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25863.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25863.4 Without the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25963.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25963.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25963.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

64 -compact space 26064.1 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26064.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26064.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26164.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26164.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 262

64.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26264.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26964.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

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

Baire space (set theory)

For the concept in topology, see Baire space.

In set theory, the Baire space is the set of all innite sequences of natural numbers with a certain topology. Thisspace is commonly used in descriptive set theory, to the extent that its elements are often called reals. It is oftendenoted B, NN, , or . Moschovakis denotes it N .The Baire space is dened to be the Cartesian product of countably innitely many copies of the set of naturalnumbers, and is given the product topology (where each copy of the set of natural numbers is given the discretetopology). The Baire space is often represented using the tree of nite sequences of natural numbers.The Baire space can be contrasted with Cantor space, the set of innite sequences of binary digits.

2.1 Topology and treesThe product topology used to dene the Baire space can be described more concretely in terms of trees. The denitionof the product topology leads to this characterization of basic open sets:

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

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

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

Thus a basic open set in the Baire space species a nite initial segment of an innite sequence of natural numbers,and all the innite sequences extending form a basic open set. This leads to a representation of the Baire space asthe set of all paths through the full tree

4 CHAPTER 2. BAIRE SPACE (SET THEORY)

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

2. It is zero-dimensional and totally disconnected.3. It is not locally compact.

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

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

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

2.4 References Kechris, Alexander S. (1994). Classical Descriptive Set Theory. Springer-Verlag. ISBN 0-387-94374-9. Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

Chapter 3

Banach space

In mathematics, more specically in functional analysis, aBanach space (pronounced [banax]) is a complete normedvector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length anddistance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a welldened limit that is within the space.Banach spaces are named after the Polish mathematician Stefan Banach, who introduced and made a systematic studyof them in 19201922 along with Hans Hahn and Eduard Helly.[1] Banach spaces originally grew out of the study offunction spaces by Hilbert, Frchet, and Riesz earlier in the century. Banach spaces play a central role in functionalanalysis. In other areas of analysis, the spaces under study are often Banach spaces.

3.1 DenitionA Banach space is a vector space X over the eld R of real numbers, or over the eld C of complex numbers, whichis equipped with a norm and which is complete with respect to that norm, that is to say, for every Cauchy sequence{xn} in X, there exists an element x in X such that

limn!1xn = x;

or equivalently:

limn!1 kxn xkX = 0:

The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series ofvectors. A normed space X is a Banach space if and only if each absolutely convergent series in X converges,[2]

1Xn=1

kvnkX

6 CHAPTER 3. BANACH SPACE

3.2.1 Linear operators, isomorphismsMain article: Bounded operator

If X and Y are normed spaces over the same ground eld K, the set of all continuous K-linear maps T : X Yis denoted by B(X, Y). In innite-dimensional spaces, not all linear maps are continuous. A linear mapping from anormed space X to another normed space is continuous if and only if it is bounded on the closed unit ball of X. Thus,the vector space B(X, Y) can be given the operator norm

kTk = sup fkTxkY j x 2 X; kxkX 1g :For Y a Banach space, the space B(X, Y) is a Banach space with respect to this norm.If X is a Banach space, the space B(X) = B(X, X) forms a unital Banach algebra; the multiplication operation is givenby the composition of linear maps.If X and Y are normed spaces, they are isomorphic normed spaces if there exists a linear bijection T : X Y suchthat T and its inverse T 1 are continuous. If one of the two spaces X or Y is complete (or reexive, separable, etc.)then so is the other space. Two normed spaces X and Y are isometrically isomorphic if in addition, T is an isometry,i.e., ||T(x)|| = ||x|| for every x in X. The Banach-Mazur distance d(X, Y) between two isomorphic but not isometricspaces X and Y gives a measure of how much the two spaces X and Y dier.

3.2.2 Basic notionsEvery normed space X can be isometrically embedded in a Banach space. More precisely, there is a Banach space Yand an isometric mapping T : X Y such that T(X) is dense in Y. If Z is another Banach space such that there is anisometric isomorphism from X onto a dense subset of Z, then Z is isometrically isomorphic to Y.This Banach space Y is the completion of the normed space X. The underlying metric space for Y is the same asthe metric completion of X, with the vector space operations extended from X to Y. The completion of X is oftendenoted by bX .The cartesian product X Y of two normed spaces is not canonically equipped with a norm. However, severalequivalent norms are commonly used,[4] such as

k(x; y)k1 = kxk+ kyk; k(x; y)k1 = max(kxk; kyk)and give rise to isomorphic normed spaces. In this sense, the product X Y (or the direct sum X Y) is complete ifand only if the two factors are complete.If M is a closed linear subspace of a normed space X, there is a natural norm on the quotient space X / M,

kx+Mk = infm2M

kx+mk:

The quotient X / M is a Banach space when X is complete.[5] The quotient map from X onto X / M, sending x in Xto its class x + M, is linear, onto and has norm 1, except when M = X, in which case the quotient is the null space.The closed linear subspace M of X is said to be a complemented subspace of X if M is the range of a boundedlinear projection P from X onto M. In this case, the space X is isomorphic to the direct sum of M and Ker(P), thekernel of the projection P.Suppose that X and Y are Banach spaces and that T B(X, Y). There exists a canonical factorization of T as[5]

T = T1 ; T : X ! X/Ker(T ) T1! Ywhere the rst map is the quotient map, and the second map T1 sends every class x + Ker(T) in the quotient to theimage T(x) in Y. This is well dened because all elements in the same class have the same image. The mapping T1is a linear bijection from X / Ker(T) onto the range T(X), whose inverse need not be bounded.

3.2. GENERAL THEORY 7

3.2.3 Classical spacesBasic examples[6] of Banach spaces include: the Lp spaces and their special cases, the sequence spaces p that consistof scalar sequences indexed by N; among them, the space 1 of absolutely summable sequences and the space 2 ofsquare summable sequences; the space c0 of sequences tending to zero and the space of bounded sequences; thespace C(K) of continuous scalar functions on a compact Hausdor space K, equipped with the max norm,

kfkC(K) = maxfjf(x)j : x 2 Kg; f 2 C(K):According to the BanachMazur theorem, every Banach space is isometrically isomorphic to a subspace of someC(K).[7] For every separable Banach space X, there is a closed subspace M of 1 such that X 1/M.[8]

Any Hilbert space serves as an example of a Banach space. A Hilbert space H on K = R, C is complete for a normof the form

kxkH =phx; xi;

where

h; i : H H ! Kis the inner product, linear in its rst argument that satises the following:

8x; y 2 H : hy; xi = hx; yi;8x 2 H : hx; xi 0;hx; xi = 0, x = 0:

For example, the space L2 is a Hilbert space.The Hardy spaces, the Sobolev spaces are examples of Banach spaces that are related to Lp spaces and have additionalstructure. They are important in dierent branches of analysis, Harmonic analysis and Partial dierential equationsamong others.

3.2.4 Banach algebrasA Banach algebra is a Banach space A over K = R or C, together with a structure of algebra over K, such that theproduct map (a, b) A A ab A is continuous. An equivalent norm on A can be found so that ||ab|| ||a|| ||b||for all a, b A.

Examples

The Banach space C(K), with the pointwise product, is a Banach algebra. The disk algebra A(D) consists of functions holomorphic in the open unit disk D C and continuous on its

closure: D. Equipped with the max norm on D, the disk algebra A(D) is a closed subalgebra of C(D). The Wiener algebra A(T) is the algebra of functions on the unit circle T with absolutely convergent Fourier

series. Via the map associating a function on T to the sequence of its Fourier coecients, this algebra isisomorphic to the Banach algebra 1(Z), where the product is the convolution of sequences.

For every Banach space X, the space B(X) of bounded linear operators on X, with the composition of maps asproduct, is a Banach algebra.

A C*-algebra is a complex Banach algebra A with an antilinear involution a a such that ||aa|| = ||a||2.The space B(H) of bounded linear operators on a Hilbert space H is a fundamental example of C*-algebra.The GelfandNaimark theorem states that every C*-algebra is isometrically isomorphic to a C*-subalgebra ofsome B(H). The space C(K) of complex continuous functions on a compact Hausdor space K is an exampleof commutative C*-algebra, where the involution associates to every function f its complex conjugate f .

8 CHAPTER 3. BANACH SPACE

3.2.5 Dual spaceMain article: Dual space

If X is a normed space and K the underlying eld (either the real or the complex numbers), the continuous dualspace is the space of continuous linear maps from X into K, or continuous linear functionals. The notation for thecontinuous dual is X = B(X, K) in this article.[9] Since K is a Banach space (using the absolute value as norm), thedual X is a Banach space, for every normed space X.The main tool for proving the existence of continuous linear functionals is the HahnBanach theorem.

HahnBanach theorem. Let X be a vector space over the eld K = R, C. Let further

Y X be a linear subspace, p : X R be a sublinear function and f : Y K be a linear functional so that Re( f (y)) p(y) for all y in Y.

Then, there exists a linear functional F : X K so that

F jY = f; and 8x 2 X; Re(F (x)) p(x):

In particular, every continuous linear functional on a subspace of a normed space can be continuously extended to thewhole space, without increasing the norm of the functional.[10] An important special case is the following: for everyvector x in a normed space X, there exists a continuous linear functional f on X such that

f(x) = kxkX ; kfkX0 1:

When x is not equal to the 0 vector, the functional f must have norm one, and is called a norming functional for x.The HahnBanach separation theorem states that two disjoint non-empty convex sets in a real Banach space, oneof them open, can be separated by a closed ane hyperplane. The open convex set lies strictly on one side of thehyperplane, the second convex set lies on the other side but may touch the hyperplane.[11]

A subset S in a Banach space X is total if the linear span of S is dense in X. The subset S is total in X if andonly if the only continuous linear functional that vanishes on S is the 0 functional: this equivalence follows from theHahnBanach theorem.If X is the direct sum of two closed linear subspaces M and N, then the dual X of X is isomorphic to the direct sumof the duals of M and N.[12] If M is a closed linear subspace in X, one can associate the orthogonal of M in the dual,

M? = fx0 2 X 0 : x0(m) = 0; 8m 2Mg :

The orthogonal M is a closed linear subspace of the dual. The dual of M is isometrically isomorphic to X / M .The dual of X / M is isometrically isomorphic to M .[13]

The dual of a separable Banach space need not be separable, but:

Theorem.[14] Let X be a normed space. If X is separable, then X is separable.

When X is separable, the above criterion for totality can be used for proving the existence of a countable total subsetin X.

Weak topologies

The weak topology on a Banach space X is the coarsest topology on X for which all elements x in the continuousdual space X are continuous. The norm topology is therefore ner than the weak topology. It follows from theHahnBanach separation theorem that the weak topology is Hausdor, and that a norm-closed convex subset of a

3.2. GENERAL THEORY 9

Banach space is also weakly closed.[15] A norm-continuous linear map between two Banach spaces X and Y is alsoweakly continuous, i.e., continuous from the weak topology of X to that of Y.[16]

If X is innite-dimensional, there exist linear maps which are not continuous. The space X of all linear maps fromX to the underlying eld K (this space X is called the algebraic dual space, to distinguish it from X ) also induces atopology on X which is ner than the weak topology, and much less used in functional analysis.On a dual space X , there is a topology weaker than the weak topology of X , called weak* topology. It is thecoarsest topology on X for which all evaluation maps x X x(x), x X, are continuous. Its importance comesfrom the BanachAlaoglu theorem.

BanachAlaoglu Theorem. Let X be a normed vector space. Then the closed unit ball B = {x X : ||x|| 1} of the dual space is compact in the weak* topology.

The BanachAlaoglu theorem depends on Tychonos theorem about innite products of compact spaces. When Xis separable, the unit ball B of the dual is a metrizable compact in the weak* topology.[17]

Examples of dual spaces

The dual of c0 is isometrically isomorphic to 1: for every bounded linear functional f on c0, there is a unique elementy = {yn} 1 such that

f(x) =Xn2N

xnyn; x = fxng 2 c0; and kfk(c0)0 = kyk`1 :

The dual of 1 is isometrically isomorphic to . The dual of Lp([0, 1]) is isometrically isomorphic to Lq([0, 1])when 1 p < and 1/p + 1/q = 1.For every vector y in a Hilbert space H, the mapping

x 2 H ! fy(x) = hx; yidenes a continuous linear functional fy on H. The Riesz representation theorem states that every continuous linearfunctional on H is of the form fy for a uniquely dened vector y in H. The mapping y H fy is an antilinearisometric bijection from H onto its dual H . When the scalars are real, this map is an isometric isomorphism.When K is a compact Hausdor topological space, the dualM(K) of C(K) is the space of Radon measures in the senseof Bourbaki.[18] The subset P(K) of M(K) consisting of non-negative measures of mass 1 (probability measures) is aconvex w*-closed subset of the unit ball of M(K). The extreme points of P(K) are the Dirac measures on K. The setof Dirac measures on K, equipped with the w*-topology, is homeomorphic to K.

Banach-Stone Theorem. If K and L are compact Hausdor spaces and if C(K) and C(L) are isomet-rically isomorphic, then the topological spaces K and L are homeomorphic.[19][20]

The result has been extended by Amir[21] and Cambern[22] to the case when the multiplicative BanachMazur distancebetween C(K) and C(L) is < 2. The theorem is no longer true when the distance is = 2.[23]

In the commutative Banach algebra C(K), the maximal ideals are precisely kernels of Dirac mesures on K,

Ix = ker x = ff 2 C(K) : f(x) = 0g; x 2 K:More generally, by the Gelfand-Mazur theorem, the maximal ideals of a unital commutative Banach algebra can beidentied with its characters---not merely as sets but as topological spaces: the former with the hull-kernel topologyand the latter with the w*-topology. In this identication, the maximal ideal space can be viewed as a w*-compactsubset of the unit ball in the dual A .

Theorem. If K is a compact Hausdor space, then the maximal ideal space of the Banach algebraC(K) is homeomorphic to K.[19]

10 CHAPTER 3. BANACH SPACE

Not every unital commutative Banach algebra is of the form C(K) for some compact Hausdor space K. However,this statement holds if one places C(K) in the smaller category of commutative C*-algebras. Gelfands representationtheorem for commutative C*-algebras states that every commutative unital C*-algebra A is isometrically isomorphicto a C(K) space.[24] The Hausdor compact space K here is again the maximal ideal space, also called the spectrumof A in the C*-algebra context.

Bidual

If X is a normed space, the (continuous) dual X of the dual X is called bidual, or second dual of X. For everynormed space X, there is a natural map,

(FX : X ! X 00FX(x)(f) = f(x) 8x 2 X;8f 2 X 0

This denes FX(x) as a continuous linear functional on X , i.e., an element of X . The map FX : x FX(x) is alinear map from X to X . As a consequence of the existence of a norming functional f for every x in X, this mapFX is isometric, thus injective.For example, the dual of X = c0 is identied with 1, and the dual of 1 is identied with , the space of boundedscalar sequences. Under these identications, FX is the inclusion map from c0 to . It is indeed isometric, but notonto.If FX is surjective, then the normed space X is called reexive (see below). Being the dual of a normed space, thebidual X is complete, therefore, every reexive normed space is a Banach space.Using the isometric embedding FX, it is customary to consider a normed space X as a subset of its bidual. When Xis a Banach space, it is viewed as a closed linear subspace of X . If X is not reexive, the unit ball of X is a propersubset of the unit ball of X . The Goldstine theorem states that the unit ball of a normed space is weakly*-dense inthe unit ball of the bidual. In other words, for every x in the bidual, there exists a net {xj} in X so that

supjkxjk kx00k; x00(f) = lim

jf(xj); f 2 X 0:

The net may be replaced by a weakly*-convergent sequence when the dual X is separable. On the other hand,elements of the bidual of 1 that are not in 1 cannot be weak*-limit of sequences in 1, since 1 is weakly sequentiallycomplete.

3.2.6 Banachs theorems

Here are the main general results about Banach spaces that go back to the time of Banachs book (Banach (1932))and are related to the Baire category theorem. According to this theorem, a complete metric space (such as a Banachspace, a Frchet space or an F-space) cannot be equal to a union of countably many closed subsets with emptyinteriors. Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is alreadyequal to one of them; a Banach space with a countable Hamel basis is nite-dimensional.

BanachSteinhaus Theorem. Let X be a Banach space and Y be a normed vector space. Suppose thatF is a collection of continuous linear operators from X to Y. The uniform boundedness principle statesthat if for all x in X we have supTF ||T(x)||Y < , then supTF ||T ||Y < .

The BanachSteinhaus theorem is not limited to Banach spaces. It can be extended for example to the case whereX is a Frchet space, provided the conclusion is modied as follows: under the same hypothesis, there exists aneighborhood U of 0 in X such that all T in F are uniformly bounded on U,

supT2F

supx2U

kT (x)kY

3.2. GENERAL THEORY 11

The Open Mapping Theorem. Let X and Y be Banach spaces and T : X Y be a continuous linearoperator. Then T is surjective if and only if T is an open map.

Corollary. Every one-to-one bounded linear operator from a Banach space onto a Banach space is anisomorphism.

The First Isomorphism Theorem for Banach spaces. Suppose that X and Y are Banach spaces andthat T B(X, Y). Suppose further that the range of T is closed in Y. Then X/ Ker(T) is isomorphic toT(X).

This result is a direct consequence of the preceding Banach isomorphism theorem and of the canonical factorizationof bounded linear maps.

Corollary. If a Banach space X is the internal direct sum of closed subspaces M1, ..., Mn, then X isisomorphic to M1 ... Mn.

This is another consequence of Banachs isomorphism theorem, applied to the continuous bijection from M1 ... Mn onto X sending (m1, ..., mn) to the sum m1 + ... + mn.

The Closed Graph Theorem. Let T : X Y be a linear mapping between Banach spaces. The graphof T is closed in X Y if and only if T is continuous.

3.2.7 ReexivityMain article: Reexive space

The normed space X is