Locally Normal Space 2

Locally normal space 2Wikipedia

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 Category of topological spaces 93.1 As a concrete category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Limits and colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Relationships to other categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Compact space 124.1 Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.3.1 Open cover denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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4.3.2 Equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3.3 Compactness of subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.4 Properties of compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4.1 Functions and compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4.2 Compact spaces and set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4.3 Ordered compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.5.1 Algebraic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Connected space 215.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.1.1 Connected components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.1.2 Disconnected spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 Path connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.4 Arc connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.5 Local connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.6 Set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.7 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.8 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.9 Stronger forms of connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.11.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.11.2 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 Continuous function 296.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.2 Real-valued continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.2.3 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.2.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.2.5 Directional and semi-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.3 Continuous functions between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.3.1 Uniform, Hlder and Lipschitz continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6.4 Continuous functions between topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.4.1 Alternative denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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6.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.4.3 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.4.4 Dening topologies via continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.5 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7 H-closed space 467.1 Examples and equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

8 Hausdor space 478.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478.2 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488.3 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488.5 Preregularity versus regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498.6 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498.7 Algebra of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508.8 Academic humour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

9 Homeomorphism 519.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

9.2.1 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539.5 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

10 If and only if 5510.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

10.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.2.3 Origin of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

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10.3 Distinction from if and only if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.4 More general usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

11 Kolmogorov space 5811.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.2 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

11.2.1 Spaces which are not T0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.2.2 Spaces which are T0 but not T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

11.3 Operating with T0 spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5911.4 The Kolmogorov quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5911.5 Removing T0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6011.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

12 Limit point 6112.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112.2 Types of limit points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112.3 Some facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

13 Locally compact space 6413.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6413.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

13.2.1 Compact Hausdor spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.2.2 Locally compact Hausdor spaces that are not compact . . . . . . . . . . . . . . . . . . . 6513.2.3 Hausdor spaces that are not locally compact . . . . . . . . . . . . . . . . . . . . . . . . 6513.2.4 Non-Hausdor examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

13.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.3.1 The point at innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.3.2 Locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

13.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6713.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

14 Locally Hausdor space 6814.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

15 Locally normal space 6915.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6915.2 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6915.3 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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15.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6915.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

16 Locally regular space 7116.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7116.2 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7116.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7116.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

17 Metric space 7217.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7217.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7217.3 Examples of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7317.4 Open and closed sets, topology and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 7417.5 Types of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

17.5.1 Complete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7417.5.2 Bounded and totally bounded spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7517.5.3 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7617.5.4 Locally compact and proper spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7617.5.5 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7617.5.6 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

17.6 Types of maps between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7617.6.1 Continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7717.6.2 Uniformly continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7717.6.3 Lipschitz-continuous maps and contractions . . . . . . . . . . . . . . . . . . . . . . . . . 7717.6.4 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7817.6.5 Quasi-isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

17.7 Notions of metric space equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7817.8 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7817.9 Distance between points and sets; Hausdor distance and Gromov metric . . . . . . . . . . . . . . 7917.10Product metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

17.10.1 Continuity of distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7917.11Quotient metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8017.12Generalizations of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

17.12.1 Metric spaces as enriched categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8017.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8117.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8117.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8217.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

18 Metrization theorem 8318.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

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18.2 Metrization theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8318.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8418.4 Examples of non-metrizable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8418.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8418.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

19 Normal space 8519.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8519.2 Examples of normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8619.3 Examples of non-normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8619.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8719.5 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8719.6 Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8719.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

20 Partially ordered set 8820.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8920.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8920.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8920.4 Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . 9020.5 Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9020.6 Strict and non-strict partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9120.7 Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9120.8 Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9120.9 Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9220.10Linear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9220.11In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9320.12Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9320.13Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9320.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9320.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9420.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9420.17External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

21 Regular space 9521.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9521.2 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9621.3 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9621.4 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9721.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

22 Separated sets 9822.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

CONTENTS vii

22.2 Relation to separation axioms and separated spaces . . . . . . . . . . . . . . . . . . . . . . . . . 9922.3 Relation to connected spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9922.4 Relation to topologically distinguishable points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9922.5 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

23 Separation axiom 10023.1 Preliminary denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10023.2 Main denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10123.3 Relationships between the axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10223.4 Other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10223.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10323.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10323.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

24 Subspace topology 10624.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10624.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10624.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10724.4 Preservation of topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10824.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10824.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

25 T1 space 10925.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10925.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10925.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11025.4 Generalisations to other kinds of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11125.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

26 Topological space 11226.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

26.1.1 Neighbourhoods denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11226.1.2 Open sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11326.1.3 Closed sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11426.1.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

26.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11426.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11426.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11526.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11626.6 Classication of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11626.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11626.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11626.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

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26.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11726.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11726.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11726.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

27 Tychono space 11927.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11927.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11927.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

27.3.1 Preservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12027.3.2 Real-valued continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12027.3.3 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12127.3.4 Compactications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12127.3.5 Uniform structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

27.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

28 Uniform space 12228.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

28.1.1 Entourage denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12228.1.2 Pseudometrics denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12328.1.3 Uniform cover denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

28.2 Topology of uniform spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12328.2.1 Uniformizable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

28.3 Uniform continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12428.4 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

28.4.1 Hausdor completion of a uniform space . . . . . . . . . . . . . . . . . . . . . . . . . . . 12528.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12528.6 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12628.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12628.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

29 Upper set 12729.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12829.2 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12829.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12829.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

30 Vacuous truth 12930.1 Scope of the concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12930.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12930.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13030.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13030.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

CONTENTS ix

30.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13030.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 131

30.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13130.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13430.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

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 setW(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

Category of topological spaces

In mathematics, the category of topological spaces, often denotedTop, is the category whose objects are topologicalspaces and whose morphisms are continuous maps or some other variant; for example, objects are often assumed tobe compactly generated. This is a category because the composition of two continuous maps is again continuous. Thestudy of Top and of properties of topological spaces using the techniques of category theory is known as categoricaltopology.N.B. Some authors use the name Top for the category with topological manifolds as objects and continuous maps asmorphisms.

3.1 As a concrete categoryLike many categories, the category Top is a concrete category (also known as a construct), meaning its objects aresets with additional structure (i.e. topologies) and its morphisms are functions preserving this structure. There is anatural forgetful functor

U : Top Set

to the category of sets which assigns to each topological space the underlying set and to each continuous map theunderlying function.The forgetful functor U has both a left adjoint

D : Set Top

which equips a given set with the discrete topology and a right adjoint

I : Set Top

which equips a given set with the indiscrete topology. Both of these functors are, in fact, right inverses to U (meaningthat UD and UI are equal to the identity functor on Set). Moreover, since any function between discrete or indiscretespaces is continuous, both of these functors give full embeddings of Set into Top.The construct Top is also ber-complete meaning that the category of all topologies on a given set X (called the berof U above X) forms a complete lattice when ordered by inclusion. The greatest element in this ber is the discretetopology on X while the least element is the indiscrete topology.The construct Top is the model of what is called a topological category. These categories are characterized by thefact that every structured source (X ! UAi)I has a unique initial lift (A! Ai)I . In Top the initial lift is obtainedby placing the initial topology on the source. Topological categories have many properties in common with Top (suchas ber-completeness, discrete and indiscrete functors, and unique lifting of limits).

9

10 CHAPTER 3. CATEGORY OF TOPOLOGICAL SPACES

3.2 Limits and colimitsThe category Top is both complete and cocomplete, which means that all small limits and colimits exist in Top. Infact, the forgetful functor U : Top Set uniquely lifts both limits and colimits and preserves them as well. Therefore,(co)limits in Top are given by placing topologies on the corresponding (co)limits in Set.Specically, if F is a diagram in Top and (L, ) is a limit of UF in Set, the corresponding limit of F in Top is obtainedby placing the initial topology on (L, ). Dually, colimits in Top are obtained by placing the nal topology on thecorresponding colimits in Set.Unlike many algebraic categories, the forgetful functor U : Top Set does not create or reect limits since therewill typically be non-universal cones in Top covering universal cones in Set.Examples of limits and colimits in Top include:

The empty set (considered as a topological space) is the initial object of Top; any singleton topological spaceis a terminal object. There are thus no zero objects in Top.

The product in Top is given by the product topology on the Cartesian product. The coproduct is given by thedisjoint union of topological spaces.

The equalizer of a pair of morphisms is given by placing the subspace topology on the set-theoretic equalizer.Dually, the coequalizer is given by placing the quotient topology on the set-theoretic coequalizer.

Direct limits and inverse limits are the set-theoretic limits with the nal topology and initial topology respec-tively.

Adjunction spaces are an example of pushouts in Top.

3.3 Other properties The monomorphisms in Top are the injective continuous maps, the epimorphisms are the surjective continuous

maps, and the isomorphisms are the homeomorphisms. The extremal monomorphisms are (up to isomorphism) the subspace embeddings. Every extremal monomor-

phism is regular. The extremal epimorphisms are (essentially) the quotient maps. Every extremal epimorphism is regular. The split monomorphisms are (essentially) the inclusions of retracts into their ambient space. The split epimorphisms are (up to isomorphism) the continuous surjective maps of a space onto one of its

retracts. There are no zero morphisms in Top, and in particular the category is not preadditive. Top is not cartesian closed (and therefore also not a topos) since it does not have exponential objects for all

spaces.

3.4 Relationships to other categories The category of pointed topological spaces Top is a coslice category over Top. The homotopy category hTop has topological spaces for objects and homotopy equivalence classes of contin-

uous maps for morphisms. This is a quotient category of Top. One can likewise form the pointed homotopycategory hTop.

Top contains the important category Haus of topological spaces with the Hausdor property as a full subcat-egory. The added structure of this subcategory allows for more epimorphisms: in fact, the epimorphisms inthis subcategory are precisely those morphisms with dense images in their codomains, so that epimorphismsneed not be surjective.

3.5. REFERENCES 11

3.5 References Herrlich, Horst: Topologische Reexionen und Coreexionen. Springer Lecture Notes in Mathematics 78

(1968).

Herrlich, Horst: Categorical topology 1971 - 1981. In: General Topology and its Relations to Modern Analysisand Algebra 5, Heldermann Verlag 1983, pp. 279 383.

Herrlich, Horst & Strecker, George E.: Categorical Topology - its origins, as examplied by the unfolding ofthe theory of topological reections and coreections before 1971. In: Handbook of the History of GeneralTopology (eds. C.E.Aull & R. Lowen), Kluwer Acad. Publ. vol 1 (1997) pp. 255 341.

Admek, Ji, Herrlich, Horst, & Strecker, George E.; (1990). Abstract and Concrete Categories (4.2MB PDF).Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition).

Chapter 4

Compact space

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

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

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

12

4.1. HISTORICAL DEVELOPMENT 13

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

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

In the 1880s, it became clear that results similar to the BolzanoWeierstrass theorem could be formulated for spacesof functions rather than just numbers or geometrical points. The idea of regarding functions as themselves pointsof a generalized space dates back to the investigations of Giulio Ascoli and Cesare Arzel.[2] The culmination oftheir investigations, the ArzelAscoli theorem, was a generalization of the BolzanoWeierstrass theorem to familiesof continuous functions, the precise conclusion of which was that it was possible to extract a uniformly convergentsequence of functions from a suitable family of functions. The uniform limit of this sequence then played preciselythe same role as Bolzanos limit point. Towards the beginning of the twentieth century, results similar to that ofArzel and Ascoli began to accumulate in the area of integral equations, as investigated by David Hilbert and ErhardSchmidt. For a certain class of Green functions coming from solutions of integral equations, Schmidt had shown thata property analogous to the ArzelAscoli theorem held in the sense of mean convergenceor convergence in whatwould later be dubbed a Hilbert space. This ultimately led to the notion of a compact operator as an oshoot of thegeneral notion of a compact space. It was Maurice Frchet who, in 1906, had distilled the essence of the BolzanoWeierstrass property and coined the term compactness to refer to this general phenomenon (he used the term alreadyin his 1904 paper[3] which led to the famous 1906 thesis) .However, a dierent notion of compactness altogether had also slowly emerged at the end of the 19th century fromthe study of the continuum, which was seen as fundamental for the rigorous formulation of analysis. In 1870, EduardHeine showed that a continuous function dened on a closed and bounded interval was in fact uniformly continuous.In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller openintervals, it was possible to select a nite number of these that also covered it. The signicance of this lemma wasrecognized by mile Borel (1895), and it was generalized to arbitrary collections of intervals by Pierre Cousin (1895)and Henri Lebesgue (1904). The HeineBorel theorem, as the result is now known, is another special propertypossessed by closed and bounded sets of real numbers.This property was signicant because it allowed for the passage from local information about a set (such as thecontinuity of a function) to global information about the set (such as the uniform continuity of a function). Thissentiment was expressed by Lebesgue (1904), who also exploited it in the development of the integral now bearinghis name. Ultimately the Russian school of point-set topology, under the direction of Pavel Alexandrov and PavelUrysohn, formulated HeineBorel compactness in a way that could be applied to the modern notion of a topologicalspace. Alexandrov & Urysohn (1929) showed that the earlier version of compactness due to Frchet, now called(relative) sequential compactness, under appropriate conditions followed from the version of compactness that wasformulated in terms of the existence of nite subcovers. It was this notion of compactness that became the dominantone, because it was not only a stronger property, but it could be formulated in a more general setting with a minimumof additional technical machinery, as it relied only on the structure of the open sets in a space.

14 CHAPTER 4. COMPACT SPACE

4.2 Basic examplesAn example of a compact space is the (closed) unit interval [0,1] of real numbers. If one chooses an innite numberof distinct points in the unit interval, then there must be some accumulation point in that interval. For instance,the odd-numbered terms of the sequence 1, 1/2, 1/3, 3/4, 1/5, 5/6, 1/7, 7/8, get arbitrarily close to 0, while theeven-numbered ones get arbitrarily close to 1. The given example sequence shows the importance of including theboundary points of the interval, since the limit points must be in the space itself an open (or half-open) interval ofthe real numbers is not compact. It is also crucial that the interval be bounded, since in the interval [0,) one couldchoose the sequence of points 0, 1, 2, 3, , of which no sub-sequence ultimately gets arbitrarily close to any givenreal number.In two dimensions, closed disks are compact since for any innite number of points sampled from a disk, some subsetof those points must get arbitrarily close either to a point within the disc, or to a point on the boundary. However, anopen disk is not compact, because a sequence of points can tend to the boundary without getting arbitrarily close toany point in the interior. Likewise, spheres are compact, but a sphere missing a point is not since a sequence of pointscan tend to the missing point, thereby not getting arbitrarily close to any point within the space. Lines and planes arenot compact, since one can take a set of equally-spaced points in any given direction without approaching any point.

4.3 DenitionsVarious denitions of compactness may apply, depending on the level of generality. A subset of Euclidean space inparticular is called compact if it is closed and bounded. This implies, by the BolzanoWeierstrass theorem, that anyinnite sequence from the set has a subsequence that converges to a point in the set. Various equivalent notions ofcompactness, such as sequential compactness and limit point compactness, can be developed in general metric spaces.In general topological spaces, however, the dierent notions of compactness are not equivalent, and the most usefulnotion of compactnessoriginally called bicompactnessis dened using covers consisting of open sets (see Opencover denition below). That this form of compactness holds for closed and bounded subsets of Euclidean space isknown as the HeineBorel theorem. Compactness, when dened in this manner, often allows one to take informationthat is known locallyin a neighbourhood of each point of the spaceand to extend it to information that holdsglobally throughout the space. An example of this phenomenon is Dirichlets theorem, to which it was originallyapplied by Heine, that a continuous function on a compact interval is uniformly continuous; here, continuity is a localproperty of the function, and uniform continuity the corresponding global property.

4.3.1 Open cover denitionFormally, a topological space X is called compact if each of its open covers has a nite subcover. Otherwise, it iscalled non-compact. Explicitly, this means that for every arbitrary collection

fUg2Aof open subsets of X such that

X =[2A

U;

there is a nite subset J of A such that

X =[i2J

Ui:

Some branches of mathematics such as algebraic geometry, typically inuenced by the French school of Bourbaki,use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are bothHausdor and quasi-compact. A compact set is sometimes referred to as a compactum, plural compacta.

4.3. DEFINITIONS 15

4.3.2 Equivalent denitions

Assuming the axiom of choice, the following are equivalent:

1. A topological space X is compact.

2. Every open cover of X has a nite subcover.

3. X has a sub-base such that every cover of the space by members of the sub-base has a nite subcover (Alexanderssub-base theorem)

4. Any collection of closed subsets of X with the nite intersection property has nonempty intersection.

5. Every net on X has a convergent subnet (see the article on nets for a proof).

6. Every lter on X has a convergent renement.

7. Every ultralter on X converges to at least one point.

8. Every innite subset of X has a complete accumulation point.[4]

Euclidean space

For any subset A of Euclidean space Rn, A is compact if and only if it is closed and bounded; this is the HeineBoreltheorem.As a Euclidean space is a metric space, the conditions in the next subsection also apply to all of its subsets. Of allof the equivalent conditions, it is in practice easiest to verify that a subset is closed and bounded, for example, for aclosed interval or closed n-ball.

Metric spaces

For any metric space (X,d), the following are equivalent:

1. (X,d) is compact.

2. (X,d) is complete and totally bounded (this is also equivalent to compactness for uniform spaces).[5]

3. (X,d) is sequentially compact; that is, every sequence in X has a convergent subsequence whose limit is in X(this is also equivalent to compactness for rst-countable uniform spaces).

4. (X,d) is limit point compact; that is, every innite subset of X has at least one limit point in X.

5. (X,d) is an image of a continuous function from the Cantor set.[6]

A compact metric space (X,d) also satises the following properties:

1. Lebesgues number lemma: For every open cover of X, there exists a number > 0 such that every subset ofX of diameter < is contained in some member of the cover.

2. (X,d) is second-countable, separable and Lindelf these three conditions are equivalent for metric spaces.The converse is not true; e.g., a countable discrete space satises these three conditions, but is not compact.

3. X is closed and bounded (as a subset of any metric space whose restricted metric is d). The converse may failfor a non-Euclidean space; e.g. the real line equipped with the discrete topology is closed and bounded but notcompact, as the collection of all singleton points of the space is an open cover which admits no nite subcover.It is complete but not totally bounded.


Characterization by continuous functions

Let X be a topological space and C(X) the ring of real continuous functions on X. For each pX, the evaluation map

evp : C(X)! Rgiven by evp(f)=f(p) is a ring homomorphism. The kernel of evp is a maximal ideal, since the residue eld C(X)/kerevp is the eld of real numbers, by the rst isomorphism theorem. A topological spaceX is pseudocompact if and onlyif every maximal ideal in C(X) has residue eld the real numbers. For completely regular spaces, this is equivalent toevery maximal ideal being the kernel of an evaluation homomorphism.[7] There are pseudocompact spaces that arenot compact, though.In general, for non-pseudocompact spaces there are always maximal ideals m in C(X) such that the residue eldC(X)/m is a (non-archimedean) hyperreal eld. The framework of non-standard analysis allows for the followingalternative characterization of compactness:[8] a topological space X is compact if and only if every point x of thenatural extension *X is innitely close to a point x0 of X (more precisely, x is contained in the monad of x0).

Hyperreal denition

A space X is compact if its natural extension *X (for example, an ultrapower) has the property that every point of *Xis innitely close to a suitable point of X X . For example, an open real interval X=(0,1) is not compact becauseits hyperreal extension *(0,1) contains innitesimals, which are innitely close to 0, which is not a point of X.

4.3.3 Compactness of subspacesA subset K of a topological space X is called compact if it is compact as a subspace. Explicitly, this means that forevery arbitrary collection

fUg2Aof open subsets of X such that

K [2A

U;

there is a nite subset J of A such that

K [i2J

Ui:

4.4 Properties of compact spaces

4.4.1 Functions and compact spacesA continuous image of a compact space is compact.[9] This implies the extreme value theorem: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum.[10] (Slightly more generally,this is true for an upper semicontinuous function.) As a sort of converse to the above statements, the pre-image of acompact space under a proper map is compact.

4.4.2 Compact spaces and set operationsA closed subset of a compact space is compact.,[11] and a nite union of compact sets is compact.

4.5. EXAMPLES 17

The product of any collection of compact spaces is compact. (Tychonos theorem, which is equivalent to the axiomof choice)Every topological space X is an open dense subspace of a compact space having at most one point more than X, bythe Alexandro one-point compactication. By the same construction, every locally compact Hausdor space X isan open dense subspace of a compact Hausdor space having at most one point more than X.

4.4.3 Ordered compact spacesA nonempty compact subset of the real numbers has a greatest element and a least element.Let X be a simply ordered set endowed with the order topology. Then X is compact if and only if X is a completelattice (i.e. all subsets have suprema and inma).[12]

4.5 Examples Any nite topological space, including the empty set, is compact. More generally, any space with a nite

topology (only nitely many open sets) is compact; this includes in particular the trivial topology. Any space carrying the conite topology is compact. Any locally compact Hausdor space can be turned into a compact space by adding a single point to it, by

means of Alexandro one-point compactication. The one-point compactication of R is homeomorphic tothe circle S1; the one-point compactication of R2 is homeomorphic to the sphere S2. Using the one-pointcompactication, one can also easily construct compact spaces which are not Hausdor, by starting with anon-Hausdor space.

The right order topology or left order topology on any bounded totally ordered set is compact. In particular,Sierpinski space is compact.

R, carrying the lower limit topology, satises the property that no uncountable set is compact. In the cocountable topology on an uncountable set, no innite set is compact. Like the previous example, the

space as a whole is not locally compact but is still Lindelf. The closed unit interval [0,1] is compact. This follows from the HeineBorel theorem. The open interval (0,1)

is not compact: the open cover

1

n; 1 1

n

for n = 3, 4, does not have a nite subcover. Similarly, the set of rational numbers in the closedinterval [0,1] is not compact: the sets of rational numbers in the intervals0;

1

1n

and

1

+

1

n; 1

cover all the rationals in [0, 1] for n = 4, 5, but this cover does not have a nite subcover. (Note thatthe sets are open in the subspace topology even though they are not open as subsets of R.)

The set R of all real numbers is not compact as there is a cover of open intervals that does not have a nitesubcover. For example, intervals (n1, n+1) , where n takes all integer values in Z, cover R but there is nonite subcover.

For every natural number n, the n-sphere is compact. Again from the HeineBorel theorem, the closed unitball of any nite-dimensional normed vector space is compact. This is not true for innite dimensions; in fact,a normed vector space is nite-dimensional if and only if its closed unit ball is compact.

On the other hand, the closed unit ball of the dual of a normed space is compact for the weak-* topology.(Alaoglus theorem)


The Cantor set is compact. In fact, every compact metric space is a continuous image of the Cantor set. Consider the set K of all functions f : R [0,1] from the real number line to the closed unit interval, and dene

a topology on K so that a sequence ffng in K converges towards f 2 K if and only if ffn(x)g convergestowards f(x) for all real numbers x. There is only one such topology; it is called the topology of pointwiseconvergence or the product topology. Then K is a compact topological space; this follows from the Tychonotheorem.

Consider the set K of all functions f : [0,1] [0,1] satisfying the Lipschitz condition |f(x) f(y)| |x y| forall x, y [0,1]. Consider on K the metric induced by the uniform distance

d(f; g) = supx2[0;1]

jf(x) g(x)j:

Then by ArzelAscoli theorem the space K is compact.

The spectrum of any bounded linear operator on a Banach space is a nonempty compact subset of the complexnumbers C. Conversely, any compact subset of C arises in this manner, as the spectrum of some boundedlinear operator. For instance, a diagonal operator on the Hilbert space `2 may have any compact nonemptysubset of C as spectrum.

4.5.1 Algebraic examples Compact groups such as an orthogonal group are compact, while groups such as a general linear group are not. Since the p-adic integers are homeomorphic to the Cantor set, they form a compact set. The spectrum of any commutative ring with the Zariski topology (that is, the set of all prime ideals) is compact,

but never Hausdor (except in trivial cases). In algebraic geometry, such topological spaces are examples ofquasi-compact schemes, quasi referring to the non-Hausdor nature of the topology.

The spectrum of a Boolean algebra is compact, a fact which is part of the Stone representation theorem. Stonespaces, compact totally disconnected Hausdor spaces, form the abstract framework in which these spectraare studied. Such spaces are also useful in the study of pronite groups.

The structure space of a commutative unital Banach algebra is a compact Hausdor space. The Hilbert cube is compact, again a consequence of Tychonos theorem. A pronite group (e.g., Galois group) is compact.

4.6 See also Compactly generated space Eberlein compactum Exhaustion by compact sets Lindelf space Metacompact space Noetherian space Orthocompact space Paracompact space

4.7. NOTES 19

4.7 Notes[1] Kline 1972, pp. 952953; Boyer & Merzbach 1991, p. 561

[2] Kline 1972, Chapter 46, 2

[3] Frechet, M. 1904. Generalisation d'un theorem de Weierstrass. Analyse Mathematique.

[4] (Kelley 1955, p. 163)

[5] Arkhangelskii & Fedorchuk 1990, Theorem 5.3.7

[6] Willard 1970 Theorem 30.7.

[7] Gillman & Jerison 1976, 5.6

[8] Robinson, Theorem 4.1.13

[9] Arkhangelskii & Fedorchuk 1990, Theorem 5.2.2; See also Compactness is preserved under a continuous map at PlanetMath.org.

[10] Arkhangelskii & Fedorchuk 1990, Corollary 5.2.1

[11] Arkhangelskii & Fedorchuk 1990, Theorem 5.2.3; Closed set in a compact space is compact at PlanetMath.org. ; Closedsubsets of a compact set are compact at PlanetMath.org.

[12] (Steen & Seebach 1995, p. 67)

4.8 References Alexandrov, Pavel; Urysohn, Pavel (1929), Mmoire sur les espaces topologiques compacts, KoninklijkeNederlandse Akademie van Wetenschappen te Amsterdam, Proceedings of the section of mathematical sciences14.

Arkhangelskii, A.V.; Fedorchuk, V.V. (1990), The basic concepts and constructions of general topology,in Arkhangelskii, A.V.; Pontrjagin, L.S., General topology I, Encyclopedia of the Mathematical Sciences 17,Springer, ISBN 978-0-387-18178-3.

Arkhangelskii, A.V. (2001), Compact space, in Hazewinkel, Michiel, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4.

Bolzano, Bernard (1817), Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein ent-gegengesetzes Resultat gewhren, wenigstens eine reele Wurzel der Gleichung liege, Wilhelm Engelmann (Purelyanalytic proof of the theorem that between any two values which give results of opposite sign, there lies at leastone real root of the equation).

Borel, mile (1895), Sur quelques points de la thorie des fonctions, Annales Scientiques de l'cole NormaleSuprieure, 3 12: 955, JFM 26.0429.03

Boyer, Carl B. (1959), The history of the calculus and its conceptual development, New York: Dover Publica-tions, MR 0124178.

Arzel, Cesare (1895), Sulle funzioni di linee, Mem. Accad. Sci. Ist. Bologna Cl. Sci. Fis. Mat. 5 (5):5574.

Arzel, Cesare (18821883), Un'osservazione intorno alle serie di funzioni, Rend. Dell' Accad. R. Delle Sci.Dell'Istituto di Bologna: 142159.

Ascoli, G. (18831884), Le curve limiti di una variet data di curve, Atti della R. Accad. Dei Lincei Memoriedella Cl. Sci. Fis. Mat. Nat. 18 (3): 521586.

Frchet, Maurice (1906), Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Matematico diPalermo 22 (1): 172, doi:10.1007/BF03018603.

Gillman, Leonard; Jerison, Meyer (1976), Rings of continuous functions, Springer-Verlag. Kelley, John (1955), General topology, Graduate Texts in Mathematics 27, Springer-Verlag.


Kline, Morris (1972), Mathematical thought from ancient to modern times (3rd ed.), Oxford University Press(published 1990), ISBN 978-0-19-506136-9.

Lebesgue, Henri (1904), Leons sur l'intgration et la recherche des fonctions primitives, Gauthier-Villars. Robinson, Abraham (1996), Non-standard analysis, Princeton University Press, ISBN 978-0-691-04490-3,

MR 0205854.

Scarborough, C.T.; Stone, A.H. (1966), Products of nearly compact spaces, Transactions of the AmericanMathematical Society (Transactions of the American Mathematical Society, Vol. 124, No. 1) 124 (1): 131147, doi:10.2307/1994440, JSTOR 1994440.

Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

Willard, Stephen (1970), General Topology, Dover publications, ISBN 0-486-43479-6

4.9 External links Countably compact at PlanetMath.org. Sundstrm, Manya Raman (2010). A pedagogical history of compactness. v1. arXiv:1006.4131 [math.HO].

This article incorporates material from Examples of compact spaces on PlanetMath, which is licensed under the CreativeCommons Attribution/Share-Alike License.

Chapter 5

Connected space

For other uses, see Connection (disambiguation).Connected and disconnected subspaces of R

A

B

C

D

E4

E1

E2

E3

From top to bottom: red space A, pink space B, yellow space C and orange space D are all connected, whereas greenspace E (made of subsets E1, E2, E3, and E4) is not connected. Furthermore, A and B are also simply connected(genus 0), while C and D are not: C has genus 1 and D has genus 4.

In topology and related branches of mathematics, a connected space is a topological space that cannot be representedas the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topologicalproperties that is used to distinguish topological spaces. A stronger notion is that of a path-connected space, whichis a space where any two points can be joined by a path.

21

22 CHAPTER 5. CONNECTED SPACE

A subset of a topological space X is a connected set if it is a connected space when viewed as a subspace of X.An example of a space that is not connected is a plane with an innite line deleted from it. Other examples ofdisconnected spaces (that is, spaces which are not connected) include the plane with an annulus removed, as well asthe union of two disjoint closed disks, where all examples of this paragraph bear the subspace topology induced bytwo-dimensional Euclidean space.

5.1 Formal denitionA topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, Xis said to be connected. A subset of a topological space is said to be connected if it is connected under its subspacetopology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article doesnot follow that practice.For a topological space X the following conditions are equivalent:

1. X is connected.

2. X cannot be divided into two disjoint nonempty closed sets.

3. The only subsets of X which are both open and closed (clopen sets) are X and the empty set.

4. The only subsets of X with empty boundary are X and the empty set.

5. X cannot be written as the union of two nonempty separated sets (sets whose closures are disjoint).

6. All continuous functions from X to {0,1} are constant, where {0,1} is the two-point space endowed with thediscrete topology.

5.1.1 Connected components

The maximal connected subsets (ordered by inclusion) of a nonempty topological space are called the connectedcomponents of the space. The components of any topological space X form a partition of X: they are disjoint,nonempty, and their union is the whole space. Every component is a closed subset of the original space. It followsthat, in the case where their number is nite, each component is also an open subset. However, if their number isinnite, this might not be the case; for instance, the connected components of the set of the rational numbers are theone-point sets, which are not open.Let x be the connected component of x in a topological space X, and 0x be the intersection of all open-closed setscontaining x (called quasi-component of x.) Then x 0x where the equality holds if X is compact Hausdor orlocally connected.

5.1.2 Disconnected spaces

A space in which all components are one-point sets is called totally disconnected. Related to this property, a space Xis called totally separated if, for any two distinct elements x and y of X, there exist disjoint open neighborhoods Uof x and V of y such that X is the union of U and V. Clearly any totally separated space is totally disconnected, butthe converse does not hold. For example take two copies of the rational numbers Q, and identify them at every pointexcept zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering thetwo copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdor, and the conditionof being totally separated is strictly stronger than the condition of being Hausdor.

5.2 Examples The closed interval [0, 2] in the standard subspace topology is connected; although it can, for example, be

written as the union of [0, 1) and [1, 2], the second set is not open in the chosen topology of [0, 2].

5.3. PATH CONNECTEDNESS 23

The union of [0, 1) and (1, 2] is disconnected; both of these intervals are open in the standard topological space[0, 1) (1, 2].

(0, 1) {3} is disconnected.

A convex set is connected; it is actually simply connected.

A Euclidean plane excluding the origin, (0, 0), is connected, but is not simply connected. The three-dimensionalEuclidean space without the origin is connected, and even simply connected. In contrast, the one-dimensionalEuclidean space without the origin is not connected.

A Euclidean plane with a straight line removed is not connected since it consists of two half-planes.

, The space of real numbers with the usual topology, is connected.

If even a single point is removed from , the remainder is disconnected. However, if even a countable innityof points are removed from n, where n2, the remainder is connected.

Any topological vector space over a connected eld is connected.

Every discrete topological space with at least two elements is disconnected, in fact such a space is totallydisconnected. The simplest example is the discrete two-point space.[1]

On the other hand, a nite set might be connected. For example, the spectrum of a discrete valuation ringconsists of two points and is connected. It is an example of a Sierpiski space.

The Cantor set is totally disconnected; since the set contains uncountably many points, it has uncountably manycomponents.

If a space X is homotopy equivalent to a connected space, then X is itself connected.

The topologists sine curve is an example of a set that is connected but is neither path connected nor locallyconnected.

The general linear group GL(n;R) (that is, the group of n-by-n real, invertible matrices) consists of two con-nected components: the one with matrices of positive determinant and the other of negative determinant.In particular, it is not connected. In contrast, GL(n;C) is connected. More generally, the set of invertiblebounded operators on a (complex) Hilbert space is connected.

The spectra of commutative local ring and integral domains are connected. More generally, the following areequivalent[2]

1. The spectrum of a commutative ring R is connected2. Every nitely generated projective module over R has constant rank.3. R has no idempotent 6= 0; 1 (i.e., R is not a product of two rings in a nontrivial way).

5.3 Path connectednessA path from a point x to a point y in a topological space X is a continuous function f from the unit interval [0,1] toX with f(0) = x and f(1) = y. A path-component of X is an equivalence class of X under the equivalence relationwhich makes x equivalent to y if there is a path from x to y. The space X is said to be path-connected (or pathwiseconnected or 0-connected) if there is exactly one path-component, i.e. if there is a path joining any two points inX. Again, many authors exclude the empty space.Every path-connected space is connected. The converse is not always true: examples of connected spaces that arenot path-connected include the extended long line L* and the topologists sine curve.However, subsets of the real lineR are connected if and only if they are path-connected; these subsets are the intervalsofR. Also, open subsets ofRn orCn are connected if and only if they are path-connected. Additionally, connectednessand path-connectedness are the same for nite topological spaces.


This subspace of R is path-connected, because a path can be drawn between any two points in the space.

5.4 Arc connectednessA space X is said to be arc-connected or arcwise connected if any two distinct points can be joined by an arc, thatis a path f which is a homeomorphism between the unit interval [0, 1] and its image f([0, 1]). It can be shown anyHausdor space which is path-connected is also arc-connected. An example of a space which is path-connected butnot arc-connected is provided by adding a second copy 0' of 0 to the nonnegative real numbers [0, ). One endowsthis set with a partial order by specifying that 0'

5.6. SET OPERATIONS 25

AB

A B

AB

A

Bconnexenon connexe

intersection intersection

connexe non connexe

union union

Examples of unions and intersections of connected sets

The union of connected sets is not necessarily connected. Consider a collection fXig of connected sets whose unionis X = [iXi . If X is disconnected and U [ V is a separation of X (with U; V disjoint and open in X ), then eachXi must be entirely contained in either U or V , since otherwise, Xi \ U and Xi \ V (which are disjoint and openin Xi ) would be a separation of Xi , contradicting the assumption that it is connected.This means that, if the union X is disconnected, then the collection fXig can be partitioned to two sub-collections,such that the unions of the sub-collections are disjoint and open in X (see picture). This implies that in several cases,a union of connected sets is necessarily connected. In particular:

1. If the common intersection of all sets is not empty ( \Xi 6= ; ), then obviously they cannot be partitioned tocollections with disjoint unions. Hence the union of connected sets with non-empty intersection is connected.

2. If the intersection of each pair of sets is not empty ( 8i; j : Xi\Xj 6= ; ) then again they cannot be partitionedto collections with disjoint unions, so their union must be connected.

3. If the sets can be ordered as a linked chain, i.e. indexed by integer indices and 8i : Xi \Xi+1 6= ; , thenagain their union must be connected.

4. If the sets are pairwise-disjoint and the quotient space X/fXig is connected, then X must be connected.Otherwise, if U [ V is a separation of X then q(U) [ q(V ) is a separation of the quotient space (sinceq(U); q(V ) are disjoint and open in the quotient space).[3]


Each ellipse is a connected set, but the union is not connected, since it can be partitioned to two disjoint open sets U and V.

Two connected sets whose dierence is not connected

The set dierence of connected sets is not necessarily connected. However, if XY and their dierence X\Y isdisconnected (and thus can be written as a union of two open sets X1 and X2), then the union of Y with each suchcomponent is connected (i.e. YXi is connected for all i). Proof:[4] By contradiction, suppose YX1 is not connected.So it can be written as the union of two disjoi

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