Locally normal space 3Wikipedia

Contents

1 Alexandrov topology 11.1 Characterizations of Alexandrov topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Duality with preordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 The Alexandrov topology on a preordered set . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 The specialization preorder on a topological space . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Equivalence between preorders and Alexandrov topologies . . . . . . . . . . . . . . . . . 21.2.4 Equivalence between monotony and continuity . . . . . . . . . . . . . . . . . . . . . . . . 31.2.5 Category theoretic description of the duality . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.6 Relationship to the construction of modal algebras from modal frames . . . . . . . . . . . 4

1.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Approach space 62.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Categorical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Baire space 93.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2.1 Modern denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.2 Historical denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Baire category theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.8 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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4 Baire space (set theory) 124.1 Topology and trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3 Relation to the real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Base (topology) 145.1 Simple properties of bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 Objects dened in terms of bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.3 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.4 Base for the closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.5 Weight and character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.5.1 Increasing chains of open sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Borel set 186.1 Generating the Borel algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2 Standard Borel spaces and Kuratowski theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.3 Non-Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.4 Alternative non-equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7 Boundary (topology) 227.1 Common denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.4 Boundary of a boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

8 Bounded set 268.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.2 Metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.3 Boundedness in topological vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.4 Boundedness in order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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9 Category of topological spaces 299.1 As a concrete category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.2 Limits and colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.3 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.4 Relationships to other categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10 Compact space 3210.1 Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.2 Basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.3 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

10.3.1 Open cover denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.3.2 Equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.3.3 Compactness of subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10.4 Properties of compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.4.1 Functions and compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.4.2 Compact spaces and set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.4.3 Ordered compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

10.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.5.1 Algebraic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

10.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

11 Connected space 4111.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

11.1.1 Connected components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.1.2 Disconnected spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.3 Path connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.4 Arc connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.5 Local connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.6 Set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.7 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4711.8 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4711.9 Stronger forms of connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

11.11.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.11.2 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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12 Continuous function 4912.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.2 Real-valued continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

12.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5212.2.3 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5512.2.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5612.2.5 Directional and semi-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

12.3 Continuous functions between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812.3.1 Uniform, Hlder and Lipschitz continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 59

12.4 Continuous functions between topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.4.1 Alternative denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.4.3 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.4.4 Dening topologies via continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . 63

12.5 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6412.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6412.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

13 Countable set 6613.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.4 Formal denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6713.5 Minimal model of set theory is countable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.6 Total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

14 H-closed space 7414.1 Examples and equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7414.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7414.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

15 Hausdor space 7515.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7515.2 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7615.3 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7615.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

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15.5 Preregularity versus regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7715.6 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7715.7 Algebra of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7815.8 Academic humour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7815.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7815.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7815.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

16 Homeomorphism 7916.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7916.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

16.2.1 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8016.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8116.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8116.5 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8216.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8216.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8216.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

17 If and only if 8317.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8317.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

17.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8317.2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8417.2.3 Origin of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

17.3 Distinction from if and only if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8417.4 More general usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8517.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8517.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8517.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

18 Image 8618.1 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8618.2 Imagery (literary term) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8618.3 Moving image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8718.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8718.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8718.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

19 Kolmogorov space 9119.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9119.2 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

19.2.1 Spaces which are not T0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

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19.2.2 Spaces which are T0 but not T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9119.3 Operating with T0 spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9219.4 The Kolmogorov quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9219.5 Removing T0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9319.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

20 Limit point 9420.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9420.2 Types of limit points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9420.3 Some facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9520.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9520.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

21 Locally compact space 9721.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9721.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

21.2.1 Compact Hausdor spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9821.2.2 Locally compact Hausdor spaces that are not compact . . . . . . . . . . . . . . . . . . . 9821.2.3 Hausdor spaces that are not locally compact . . . . . . . . . . . . . . . . . . . . . . . . 9821.2.4 Non-Hausdor examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

21.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9921.3.1 The point at innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9921.3.2 Locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

21.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10021.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

22 Locally Hausdor space 10122.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

23 Locally normal space 10223.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10223.2 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10223.3 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10223.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10223.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

24 Locally regular space 10424.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10424.2 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10424.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10424.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

25 Metric space 10525.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

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25.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10525.3 Examples of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10625.4 Open and closed sets, topology and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 10725.5 Types of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

25.5.1 Complete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10725.5.2 Bounded and totally bounded spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10825.5.3 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10925.5.4 Locally compact and proper spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10925.5.5 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10925.5.6 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

25.6 Types of maps between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10925.6.1 Continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11025.6.2 Uniformly continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11025.6.3 Lipschitz-continuous maps and contractions . . . . . . . . . . . . . . . . . . . . . . . . . 11025.6.4 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11125.6.5 Quasi-isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

25.7 Notions of metric space equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11125.8 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11125.9 Distance between points and sets; Hausdor distance and Gromov metric . . . . . . . . . . . . . . 11225.10Product metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

25.10.1 Continuity of distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11225.11Quotient metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11325.12Generalizations of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

25.12.1 Metric spaces as enriched categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11325.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11425.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11425.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11525.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

26 Metrization theorem 11626.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11626.2 Metrization theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11626.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11726.4 Examples of non-metrizable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11726.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11726.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

27 Normal space 11827.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11827.2 Examples of normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11927.3 Examples of non-normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11927.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

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27.5 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12027.6 Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12027.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

28 Partially ordered set 12128.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12228.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12228.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12228.4 Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . 12328.5 Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12328.6 Strict and non-strict partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12428.7 Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12428.8 Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12428.9 Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12528.10Linear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12528.11In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12628.12Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12628.13Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12628.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12628.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12728.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12728.17External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

29 Regular space 12829.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12829.2 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12929.3 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12929.4 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13029.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

30 Separated sets 13130.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13130.2 Relation to separation axioms and separated spaces . . . . . . . . . . . . . . . . . . . . . . . . . 13230.3 Relation to connected spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13230.4 Relation to topologically distinguishable points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13230.5 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

31 Separation axiom 13331.1 Preliminary denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13331.2 Main denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13431.3 Relationships between the axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13531.4 Other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13531.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

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31.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13631.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

32 Sigma-algebra 13932.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

32.1.1 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13932.1.2 Limits of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14032.1.3 Sub -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

32.2 Denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14132.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14132.2.2 Dynkins - theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14132.2.3 Combining -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14132.2.4 -algebras for subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14232.2.5 Relation to -ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14232.2.6 Typographic note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

32.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14332.3.1 Simple set-based examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14332.3.2 Stopping time sigma-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

32.4 -algebras generated by families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14332.4.1 -algebra generated by an arbitrary family . . . . . . . . . . . . . . . . . . . . . . . . . . 14332.4.2 -algebra generated by a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14332.4.3 Borel and Lebesgue -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14432.4.4 Product -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14432.4.5 -algebra generated by cylinder sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14432.4.6 -algebra generated by random variable or vector . . . . . . . . . . . . . . . . . . . . . . 14532.4.7 -algebra generated by a stochastic process . . . . . . . . . . . . . . . . . . . . . . . . . . 145

32.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14532.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14632.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

33 Subspace topology 14733.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14733.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14733.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14833.4 Preservation of topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14933.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14933.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

34 T1 space 15034.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15034.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15034.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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34.4 Generalisations to other kinds of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15234.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

35 Topological space 15335.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

35.1.1 Neighbourhoods denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15335.1.2 Open sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15435.1.3 Closed sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15535.1.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

35.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15535.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15535.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15635.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15735.6 Classication of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15735.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15735.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15735.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15735.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15835.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15835.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15835.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

36 Tychono space 16036.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16036.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16036.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

36.3.1 Preservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16136.3.2 Real-valued continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16136.3.3 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16236.3.4 Compactications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16236.3.5 Uniform structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

36.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

37 Uniform space 16337.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

37.1.1 Entourage denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16337.1.2 Pseudometrics denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16437.1.3 Uniform cover denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

37.2 Topology of uniform spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16437.2.1 Uniformizable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

37.3 Uniform continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16537.4 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

CONTENTS xi

37.4.1 Hausdor completion of a uniform space . . . . . . . . . . . . . . . . . . . . . . . . . . . 16637.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16637.6 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16737.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16737.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

38 Upper set 16838.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16938.2 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16938.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16938.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

39 Vacuous truth 17039.1 Scope of the concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17039.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17039.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17139.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17139.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17139.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17139.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 172

39.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17239.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17739.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Chapter 1

Alexandrov topology

In topology, an Alexandrov space (or Alexandrov-discrete space) is a topological space in which the intersectionof any family of open sets is open. It is an axiom of topology that the intersection of any nite family of open sets isopen. In an Alexandrov space the nite restriction is dropped.Alexandrov topologies are uniquely determined by their specialization preorders. Indeed, given any preorder on aset X, there is a unique Alexandrov topology on X for which the specialization preorder is . The open sets are justthe upper sets with respect to . Thus, Alexandrov topologies on X are in one-to-one correspondence with preorderson X.Alexandrov spaces are also called nitely generated spaces since their topology is uniquely determined by the familyof all nite subspaces. Alexandrov spaces can be viewed as a generalization of nite topological spaces.

1.1 Characterizations of Alexandrov topologiesAlexandrov topologies have numerous characterizations. Let X = be a topological space. Then the followingare equivalent:

Open and closed set characterizations: Open set. An arbitrary intersection of open sets in X is open. Closed set. An arbitrary union of closed sets in X is closed.

Neighbourhood characterizations: Smallest neighbourhood. Every point of X has a smallest neighbourhood. Neighbourhood lter. The neighbourhood lter of every point in X is closed under arbitrary intersec-

tions.

Interior and closure algebraic characterizations: Interior operator. The interior operator of X distributes over arbitrary intersections of subsets. Closure operator. The closure operator of X distributes over arbitrary unions of subsets.

Preorder characterizations: Specialization preorder. T is the nest topology consistent with the specialization preorder of X i.e.

the nest topology giving the preorder satisfying x y if and only if x is in the closure of {y} in X. Open up-set. There is a preorder such that the open sets of X are precisely those that are upwardly

closed i.e. if x is in the set and x y then y is in the set. (This preorder will be precisely the specializationpreorder.)

1

2 CHAPTER 1. ALEXANDROV TOPOLOGY

Closed down-set. There is a preorder such that the closed sets of X are precisely those that aredownwardly closed i.e. if x is in the set and y x then y is in the set. (This preorder will be precisely thespecialization preorder.)

Upward interior. A point x lies in the interior of a subset S of X if and only if there is a point y in Ssuch that y x where is the specialization preorder i.e. y lies in the closure of {x}.

Downward closure. A point x lies in the closure of a subset S of X if and only if there is a point y in Ssuch that x y where is the specialization preorder i.e. x lies in the closure of {y}.

Finite generation and category theoretic characterizations: Finite closure. A point x lies within the closure of a subset S of X if and only if there is a nite subsetF of S such that x lies in the closure of F.

Finite subspace. T is coherent with the nite subspaces of X. Finite inclusion map. The inclusion maps fi : Xi X of the nite subspaces of X form a nal sink. Finite generation. X is nitely generated i.e. it is in the nal hull of the nite spaces. (This means that

there is a nal sink fi : Xi X where each Xi is a nite topological space.)

Topological spaces satisfying the above equivalent characterizations are called nitely generated spaces or Alexan-drov spaces and their topology T is called the Alexandrov topology, named after the Russian mathematician PavelAlexandrov who rst investigated them.

1.2 Duality with preordered sets

1.2.1 The Alexandrov topology on a preordered setGiven a preordered set X = hX;i we can dene an Alexandrov topology on X by choosing the open sets to bethe upper sets:

= fG X : 8x; y 2 X x 2 G ^ x y ! y 2 G; gWe thus obtain a topological space T(X) = hX; i .The corresponding closed sets are the lower sets:

fS X : 8x; y 2 X x 2 S ^ y x ! y 2 S; g

1.2.2 The specialization preorder on a topological spaceGiven a topological space X = the specialization preorder on X is dened by:

xy if and only if x is in the closure of {y}.

We thus obtain a preordered set W(X) = .

1.2.3 Equivalence between preorders and Alexandrov topologiesFor every preordered set X = we always have W(T(X)) = X, i.e. the preorder of X is recovered from thetopological space T(X) as the specialization preorder. Moreover for every Alexandrov space X, we have T(W(X)) =X, i.e. the Alexandrov topology of X is recovered as the topology induced by the specialization preorder.However for a topological space in general we do not have T(W(X)) = X. Rather T(W(X)) will be the set X with aner topology than that of X (i.e. it will have more open sets).

1.2. DUALITY WITH PREORDERED SETS 3

1.2.4 Equivalence between monotony and continuityGiven a monotone function

f : XY

between two preordered sets (i.e. a function

f : XY

between the underlying sets such that xy in X implies f(x)f(y) in Y), let

T(f) : T(X)T(Y)

be the same map as f considered as a map between the corresponding Alexandrov spaces. Then

T(f) : T(X)T(Y)

is a continuous map.Conversely given a continuous map

f : XY

between two topological spaces, let

W(f) : W(X)W(Y)

be the same map as f considered as a map between the corresponding preordered sets. Then

W(f) : W(X)W(Y)

is a monotone function.Thus a map between two preordered sets is monotone if and only if it is a continuous map between the correspondingAlexandrov spaces. Conversely a map between two Alexandrov spaces is continuous if and only if it is a monotonefunction between the corresponding preordered sets.Notice however that in the case of topologies other than the Alexandrov topology, we can have a map between twotopological spaces that is not continuous but which is nevertheless still a monotone function between the correspondingpreordered sets. (To see this consider a non-Alexandrov space X and consider the identity map

i : XT(W(X)).)

1.2.5 Category theoretic description of the dualityLet Set denote the category of sets and maps. Let Top denote the category of topological spaces and continuousmaps; and let Pro denote the category of preordered sets and monotone functions. Then

T : ProTop and

W : TopPro

are concrete functors over Set which are left and right adjoints respectively.Let Alx denote the full subcategory of Top consisting of the Alexandrov spaces. Then the restrictions

4 CHAPTER 1. ALEXANDROV TOPOLOGY

T : ProAlx and

W : AlxPro

are inverse concrete isomorphisms over Set.Alx is in fact a bico-reective subcategory of Top with bico-reector TW : TopAlx. This means that given atopological space X, the identity map

i : T(W(X))X

is continuous and for every continuous map

f : YX

where Y is an Alexandrov space, the composition

i 1f : YT(W(X))

is continuous.

1.2.6 Relationship to the construction of modal algebras from modal framesGiven a preordered set X, the interior operator and closure operator of T(X) are given by:

Int(S) = { x X : for all y X, xy implies y S }, for all S X

Cl(S) = { x X : there exists a y S with xy } for all S X

Considering the interior operator and closure operator to be modal operators on the power set Boolean algebra of X,this construction is a special case of the construction of a modal algebra from a modal frame i.e. a set with a singlebinary relation. (The latter construction is itself a special case of a more general construction of a complex algebrafrom a relational structure i.e. a set with relations dened on it.) The class of modal algebras that we obtain in thecase of a preordered set is the class of interior algebrasthe algebraic abstractions of topological spaces.

1.3 HistoryAlexandrov spaces were rst introduced in 1937 by P. S. Alexandrov under the name discrete spaces, where heprovided the characterizations in terms of sets and neighbourhoods.[1] The name discrete spaces later came to be usedfor topological spaces in which every subset is open and the original concept lay forgotten. With the advancement ofcategorical topology in the 1980s, Alexandrov spaces were rediscovered when the concept of nite generation wasapplied to general topology and the name nitely generated spaces was adopted for them. Alexandrov spaces werealso rediscovered around the same time in the context of topologies resulting from denotational semantics and domaintheory in computer science.In 1966 Michael C. McCord and A. K. Steiner each independently observed a duality between partially ordered setsand spaces which were precisely the T0 versions of the spaces that Alexandrov had introduced.[2][3] P. Johnstonereferred to such topologies as Alexandrov topologies.[4] F. G. Arenas independently proposed this name for thegeneral version of these topologies.[5] McCord also showed that these spaces are weak homotopy equivalent to theorder complex of the corresponding partially ordered set. Steiner demonstrated that the duality is a contravariantlattice isomorphism preserving arbitrary meets and joins as well as complementation.It was also a well known result in the eld of modal logic that a duality exists between nite topological spaces andpreorders on nite sets (the nite modal frames for the modal logic S4). C. Naturman extended these results to aduality between Alexandrov spaces and preorders in general, providing the preorder characterizations as well as theinterior and closure algebraic characterizations.[6]

A systematic investigation of these spaces from the point of view of general topology which had been neglected sincethe original paper by Alexandrov, was taken up by F.G. Arenas.[5]

1.4. SEE ALSO 5

1.4 See also P-space, a space satisfying the weaker condition that countable intersections of open sets are open

1.5 References[1] Alexandro, P. (1937). Diskrete Rume. Mat. Sb. (N.S.) (in German) 2: 501518.

[2] McCord, M. C. (1966). Singular homology and homotopy groups of nite topological spaces. DukeMathematical Journal33 (3): 465474. doi:10.1215/S0012-7094-66-03352-7.

[3] Steiner, A. K. (1966). The Lattice of Topologies: Structure and Complementation. Transactions of the American Math-ematical Society 122 (2): 379398. doi:10.2307/1994555. ISSN 0002-9947.

[4] Johnstone, P. T. (1986). Stone spaces (1st paperback ed.). New York: Cambridge University Press. ISBN 0-521-33779-8.

[5] Arenas, F. G. (1999). Alexandro spaces (PDF). Acta Math. Univ. Comenianae 68 (1): 1725.

[6] Naturman, C. A. (1991). Interior Algebras and Topology. Ph.D. thesis, University of Cape Town Department of Mathe-matics.

Chapter 2

Approach space

In topology, approach spaces are a generalization of metric spaces, based on point-to-set distances, instead of point-to-point distances. They were introduced by Robert Lowen in 1989.

2.1 DenitionGiven a metric space (X,d), or more generally, an extended pseudoquasimetric (which will be abbreviated pq-metrichere), one can dene an induced map d:XP(X)[0,] by d(x,A) = inf { d(x,a ) : a A }. With this example inmind, a distance on X is dened to be a map XP(X)[0,] satisfying for all x in X and A, B X,

1. d(x,{x}) = 0 ;

2. d(x,) = ;

3. d(x,AB) = min d(x,A),d(x,B) ;

4. For all , 0, d(x,A) d(x,A()) + ;

where A() = { x : d(x,A) } by denition.(The empty inmum is positive innity convention is like the nullary intersection is everything convention.)An approach space is dened to be a pair (X,d) where d is a distance function on X. Every approach space has atopology, given by treating A A(0) as a Kuratowski closure operator.The appropriate maps between approach spaces are the contractions. A map f:(X,d)(Y,e) is a contraction ife(f(x),f[A]) d(x,A) for all x X, A X.

2.2 ExamplesEvery pq-metric space (X,d) can be distancized to (X,d), as described at the beginning of the denition.Given a set X, the discrete distance is given by d(x,A) = 0 if x A and = if x A. The induced topology is thediscrete topology.Given a set X, the indiscrete distance is given by d(x,A) = 0 if A is non-empty, and = if A is empty. The inducedtopology is the indiscrete topology.Given a topological space X, a topological distance is given by d(x,A) = 0 if x A, and = if not. The inducedtopology is the original topology. In fact, the only two-valued distances are the topological distances.Let P=[0,], the extended positive reals. Let d+(x,A) = max (xsup A,0) for xP and AP. Given any approachspace (X,d), the maps (for each AX) d(.,A) : (X,d) (P,d+) are contractions.

6

2.3. EQUIVALENT DEFINITIONS 7

On P, let e(x,A) = inf { |xa| : aA } for x0, NA[] .

Given a distance d, the associated AA() is a tower. Conversely, given a tower, the map d(x,A) = inf { : x A[]} is a distance, and these two operations are inverses of each other.A contraction f:(X,d)(Y,e) is, in terms of associated towers, a map such that for all 0, f[A[]] f[A][].

2.4 Categorical propertiesThe main interest in approach spaces and their contractions is that they form a category with good properties, whilestill being quantitative like metric spaces. One can take arbitrary products and coproducts and quotients, and theresults appropriately generalize the corresponding results for topologies. One can even distancize such badly non-metrizable spaces like N, the Stoneech compactication of the integers.Certain hyperspaces, measure spaces, and probabilistic metric spaces turn out to be naturally endowed with a distance.Applications have also been made to approximation theory.

2.5 References Lowen, Robert (1997). Approach spaces: the missing link in the topology-uniformity-metric triad. Oxford

Mathematical Monographs. Oxford: Clarendon Press. ISBN 0-19-850030-0. Zbl 0891.54001. Lowen, Robert (2015). Index Analysis: Approach Theory at Work. Springer.

8 CHAPTER 2. APPROACH SPACE

2.6 External links Robert Lowen

Chapter 3

Baire space

For the concept in set theory, see Baire space (set theory).

In mathematics, a Baire space is a topological space that has enough points that every intersection of a countablecollection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdor spacesare examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of Ren-LouisBaire who introduced the concept.

3.1 MotivationIn an arbitrary topological space, the class of closed sets with empty interior consists precisely of the boundaries ofdense open sets. These sets are, in a certain sense, negligible. Some examples are nite sets in , smooth curves inthe plane, and proper ane subspaces in a Euclidean space. If a topological space is a Baire space then it is large,meaning that it is not a countable union of negligible subsets. For example, the three-dimensional Euclidean space isnot a countable union of its ane planes.

3.2 DenitionThe precise denition of a Baire space has undergone slight changes throughout history, mostly due to prevailingneeds and viewpoints. First, we give the usual modern denition, and then we give a historical denition which iscloser to the denition originally given by Baire.

3.2.1 Modern denitionA Baire space is a topological space in which the union of every countable collection of closed sets with emptyinterior has empty interior.This denition is equivalent to each of the following conditions:

Every intersection of countably many dense open sets is dense. The interior of every union of countably many closed nowhere dense sets is empty. Whenever the union of countably many closed subsets of X has an interior point, then one of the closed subsets

must have an interior point.

3.2.2 Historical denitionMain article: Meagre set

9

10 CHAPTER 3. BAIRE SPACE

In his original denition, Baire dened a notion of category (unrelated to category theory) as follows.A subset of a topological space X is called

nowhere dense in X if the interior of its closure is empty of rst category or meagre in X if it is a union of countably many nowhere dense subsets of second category or nonmeagre in X if it is not of rst category in X

The denition for a Baire space can then be stated as follows: a topological spaceX is a Baire space if every non-emptyopen set is of second category in X. This denition is equivalent to the modern denition.A subset A of X is comeagre if its complement X nA is meagre. A topological space X is a Baire space if and onlyif every comeager subset of X is dense.

3.3 Examples The space R of real numbers with the usual topology, is a Baire space, and so is of second category in itself.

The rational numbers are of rst category and the irrational numbers are of second category in R . The Cantor set is a Baire space, and so is of second category in itself, but it is of rst category in the interval[0; 1] with the usual topology.

Here is an example of a set of second category in R with Lebesgue measure 0.

1\m=1

1[n=1

rn 1

2n+m; rn +

1

2n+m

where frng1n=1 is a sequence that enumerates the rational numbers.

Note that the space of rational numbers with the usual topology inherited from the reals is not a Baire space,since it is the union of countably many closed sets without interior, the singletons.

3.4 Baire category theoremMain article: Baire category theorem

The Baire category theorem gives sucient conditions for a topological space to be a Baire space. It is an importanttool in topology and functional analysis.

(BCT1) Every complete metric space is a Baire space. More generally, every topological space which ishomeomorphic to an open subset of a complete pseudometric space is a Baire space. In particular, everycompletely metrizable space is a Baire space.

(BCT2) Every locally compact Hausdor space (or more generally every locally compact sober space) is aBaire space.

BCT1 shows that each of the following is a Baire space:

The space R of real numbers The space of irrational numbers, which is homeomorphic to the Baire space of set theory The Cantor set Indeed, every Polish space

BCT2 shows that every manifold is a Baire space, even if it is not paracompact, and hence not metrizable. Forexample, the long line is of second category.

3.5. PROPERTIES 11

3.5 Properties Every non-empty Baire space is of second category in itself, and every intersection of countably many dense

open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topologicaldisjoint sum of the rationals and the unit interval [0, 1].

Every open subspace of a Baire space is a Baire space.

Given a family of continuous functions fn:XY with pointwise limit f:XY. If X is a Baire space then thepoints where f is not continuous is a meagre set in X and the set of points where f is continuous is dense in X.A special case of this is the uniform boundedness principle.

A closed subset of a Baire space is not necessarily Baire.

The product of two Baire spaces is not necessarily Baire. However, there exist sucient conditions that willguarantee that a product of arbitrarily many Baire spaces is again Baire.

3.6 See also BanachMazur game Descriptive set theory Baire space (set theory)

3.7 References

3.8 Sources Munkres, James, Topology, 2nd edition, Prentice Hall, 2000. Baire, Ren-Louis (1899), Sur les fonctions de variables relles, Annali di Mat. Ser. 3 3, 1123.

3.9 External links Encyclopaedia of Mathematics article on Baire space Encyclopaedia of Mathematics article on Baire theorem

Chapter 4

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.

4.1 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

4.3. RELATION TO THE REAL LINE 13

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.

4.3 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).

4.4 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 5

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.

5.1 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.

14

5.2. OBJECTS DEFINED IN TERMS OF BASES 15

5.2 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.

5.3 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.

5.4 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.

16 CHAPTER 5. BASE (TOPOLOGY)

5.5 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.)

5.5.1 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[

5.6. SEE ALSO 17

For

x 2 V n[

Chapter 6

Borel set

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

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

T be all countable unions of elements of T T be all countable intersections of elements of T T = (T):

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

For the base case of the denition, let G0 be the collection of open subsets of X. If i is not a limit ordinal, then i has an immediately preceding ordinal i 1. Let

Gi = [Gi1]:

If i is a limit ordinal, set

Gi =[j

6.2. STANDARD BOREL SPACES AND KURATOWSKI THEOREMS 19

G 7! G:to the rst uncountable ordinal.To prove this claim, note that any open set in a metric space is the union of an increasing sequence of closed sets.In particular, complementation of sets maps Gm into itself for any limit ordinal m; moreover if m is an uncountablelimit ordinal, Gm is closed under countable unions.Note that for each Borel set B, there is some countable ordinal B such that B can be obtained by iterating theoperation over B. However, as B varies over all Borel sets, B will vary over all the countable ordinals, and thus therst ordinal at which all the Borel sets are obtained is 1, the rst uncountable ordinal.

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

@1 2@0 = 2@0 :

6.2 Standard Borel spaces and Kuratowski theoremsLet X be a topological space. The Borel space associated to X is the pair (X,B), where B is the -algebra of Borelsets of X.Mackey dened a Borel space somewhat dierently, writing that it is a set together with a distinguished -eld ofsubsets called its Borel sets. [1] However, modern usage is to call the distinguished sub-algebra measurable sets andsuch spaces measurable spaces. The reason for this distinction is that the Borel sets are the -algebra generated byopen sets (of a topological space), whereas Mackeys denition refers to a set equipped with an arbitrary -algebra.There exist measurable spaces that are not Borel spaces, for any choice of topology on the underlying space.[2]

Measurable spaces form a category in which the morphisms are measurable functions between measurable spaces. Afunction f : X ! Y is measurable if it pulls back measurable sets, i.e., for all measurable sets B in Y, f1(B) is ameasurable set in X.Theorem. Let X be a Polish space, that is, a topological space such that there is a metric d on X which denes thetopology of X and which makes X a complete separable metric space. Then X as a Borel space is isomorphic to oneof (1) R, (2) Z or (3) a nite space. (This result is reminiscent of Maharams theorem.)Considered as Borel spaces, the real line R, the union of R with a countable set, and Rn are isomorphic.A standard Borel space is the Borel space associated to a Polish space. A standard Borel space is characterized upto isomorphism by its cardinality,[3] and any uncountable standard Borel space has the cardinality of the continuum.For subsets of Polish spaces, Borel sets can be characterized as those sets which are the ranges of continuous injectivemaps dened on Polish spaces. Note however, that the range of a continuous noninjective map may fail to be Borel.See analytic set.Every probability measure on a standard Borel space turns it into a standard probability space.

6.3 Non-Borel setsAn example of a subset of the reals which is non-Borel, due to Lusin[4] (see Sect. 62, pages 7678), is describedbelow. In contrast, an example of a non-measurable set cannot be exhibited, though its existence can be proved.

20 CHAPTER 6. BOREL SET

Every irrational number has a unique representation by a continued fraction

x = a0 +1

a1 +1

a2 +1

a3 +1

. . .where a0 is some integer and all the other numbers ak are positive integers. Let A be the set of all irrationalnumbers that correspond to sequences (a0; a1; : : : ) with the following property: there exists an innite subsequence(ak0 ; ak1 ; : : : ) such that each element is a divisor of the next element. This set A is not Borel. In fact, it is analytic,and complete in the class of analytic sets. For more details see descriptive set theory and the book by Kechris,especially Exercise (27.2) on page 209, Denition (22.9) on page 169, and Exercise (3.4)(ii) on page 14.Another non-Borel set is an inverse image f1[0] of an innite parity function f : f0; 1g! ! f0; 1g . However, thisis a proof of existence (via the axiom of choice), not an explicit example.

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

6.5 See also Baire set Cylindrical -algebra Polish space Descriptive set theory Borel hierarchy

6.6 References William Arveson, An Invitation to C*-algebras, Springer-Verlag, 1981. (See Chapter 3 for an excellent expo-

sition of Polish topology)

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

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

Halsey Royden, Real Analysis, Prentice Hall, 1988

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

6.7. NOTES 21

6.7 Notes[1] Mackey, G.W. (1966), Ergodic Theory and Virtual Groups, Math. Annalen. (Springer-Verlag) 166 (3): 187207,

doi:10.1007/BF01361167, ISSN 0025-5831, (subscription required (help))

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

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

[4] Lusin, Nicolas (1927), Sur les ensembles analytiques, Fundamenta Mathematicae (Institute of mathematics, Polishacademy of sciences) 10: 195.

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

6.8 External links Hazewinkel, Michiel, ed. (2001), Borel set, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-

010-4

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

Weisstein, Eric W., Borel Set, MathWorld.

Chapter 7

Boundary (topology)

This article is about boundaries in general topology. For the boundary of a manifold, see boundary of a manifold.In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points

A set (in light blue) and its boundary (in dark blue).

which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closureof S, not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. Notationsused for boundary of a set S include bd(S), fr(S), and S. Some authors (for example Willard, in General Topology)use the term frontier, instead of boundary in an attempt to avoid confusion with the concept of boundary used inalgebraic topology and manifold theory. However, frontier sometimes refers to a dierent set, which is the set ofboundary points which are not actually in the set; that is, S \ S.A connected component of the boundary of S is called a boundary component of S.

22

7.1. COMMON DEFINITIONS 23

7.1 Common denitionsThere are several common (and equivalent) denitions to the boundary of a subset S of a topological space X:

the closure of S without the interior of S: S = S \ So. the intersection of the closure of S with the closure of its complement: S = S (X \ S). the set of points p of X such that every neighborhood of p contains at least one point of S and at least one point

not of S.

7.2 Examples

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1.5 -1 -0.5 0 0.5 1

c.im

c.re

Boundaries of 53 hyperbolic components of Mandelbrot set made in 13sec

one period 1 component = {c:c=(2*w-w*w)/4} one period 2 component = {c:c=(w/4 -1)}

three period 3 components (blue)six period 4 components (magenta)fifteen period 5 components (black)

27 period 6 components (black)

Boundary of hyperbolic components of Mandelbrot set

Consider the real line R with the usual topology (i.e. the topology whose basis sets are open intervals). One has

(0,5) = [0,5) = (0,5] = [0,5] = {0,5} =

24 CHAPTER 7. BOUNDARY (TOPOLOGY)

Q = R (Q [0,1]) = [0,1]

These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure.In the space of rational numbers with the usual topology (the subspace topology of R), the boundary of (1; a) ,where a is irrational, is empty.The boundary of a set is a topological notion and may change if one changes the topology. For example, given theusual topology on R2, the boundary of a closed disk = {(x,y) | x2 + y2 1} is the disks surrounding circle: ={(x,y) | x2 + y2 = 1}. If the disk is viewed as a set in R3 with its own usual topology, i.e. = {(x,y,0) | x2 + y2 1},then the boundary of the disk is the disk itself: = . If the disk is viewed as its own topological space (with thesubspace topology of R2), then the boundary of the disk is empty.

7.3 Properties The boundary of a set is closed.[1]

The boundary of the interior of a set as well as the boundary of the closure of a set are both contained in theboundary of the set.

A set is the boundary of some open set if and only if it is closed and nowhere dense. The boundary of a set is the boundary of the complement of the set: S = (SC).

Hence:

p is a boundary point of a set if and only if every neighborhood of p contains at least one point in the set andat least one point not in the set.

A set is closed if and only if it contains its boundary, and open if and only if it is disjoint from its boundary. The closure of a set equals the union of the set with its boundary. S = S S. The boundary of a set is empty if and only if the set is both closed and open (that is, a clopen set).

7.4. BOUNDARY OF A BOUNDARY 25

Conceptual Venn diagram showing the relationships among dierent points of a subset S of Rn. A = setof limit points of S, B = set of boundary points of S, area shaded green = set of interior points of S, areashaded yellow = set of isolated points of S, areas shaded black = empty sets. Every point of S is either aninterior point or a boundary point. Also, every point of S is either an accumulation point or an isolatedpoint. Likewise, every boundary point of S is either an accumulation point or an isolated point. Isolatedpoints are always boundary points.

7.4 Boundary of a boundaryFor any set S, S S, with equality holding if and only if the boundary of S has no interior points, which will bethe case for example if S is either closed or open. Since the boundary of a set is closed, S = S for any set S.The boundary operator thus satises a weakened kind of idempotence.In discussing boundaries of manifolds or simplexes and their simplicial complexes, one often meets the assertion thatthe boundary of the boundary is always empty. Indeed, the construction of the singular homology rests critically onthis fact. The explanation for the apparent incongruity is that the topological boundary (the subject of this article) isa slightly dierent concept than the boundary of a manifold or of a simplicial complex. For example, the boundaryof an open disk viewed as a manifold is empty, while its boundary in the sense of topological space is the circlesurrounding the disk.

7.5 See also See the discussion of boundary in topological manifold for more details. Lebesgues density theorem, for measure-theoretic characterization and properties of boundary bounding point

7.6 References[1] Mendelson, Bert (1990) [1975]. Introduction to Topology (Third ed.). Dover. p. 86. ISBN 0-486-66352-3. Corollary 4.15

For each subset A, Brdy (A) is closed.

Munkres, J. R. (2000). Topology. Prentice-Hall. ISBN 0-13-181629-2. Willard, S. (1970). General Topology. Addison-Wesley. ISBN 0-201-08707-3. van den Dries, L. (1998). Tame Topology. ISBN 978-0521598385.

Chapter 8

Bounded set

Bounded and boundary are distinct concepts; for the latter see boundary (topology). A circle in isolationis a boundaryless bounded set, while the half plane is unbounded yet has a boundary.

In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of nitesize. Conversely, a set which is not bounded is called unbounded. The word bounded makes no sense in a generaltopological space, without a metric.

8.1 Denition

A set S of real numbers is called bounded from above if there is a real number k such that k s for all s in S. Thenumber k is called an upper bound of S. The terms bounded from below