locally normal space 1

Locally normal space 1Wikipedia

Contents

1 Compact space 11.1 Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Open cover denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.2 Equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.3 Compactness of subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Properties of compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4.1 Functions and compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4.2 Compact spaces and set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4.3 Ordered compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5.1 Algebraic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Connected space 102.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Connected components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.2 Disconnected spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Path connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Arc connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Local connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.7 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.8 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.9 Stronger forms of connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.11.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

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2.11.2 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Continuous function 183.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Real-valued continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.3 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.5 Directional and semi-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Continuous functions between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.1 Uniform, Hlder and Lipschitz continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Continuous functions between topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4.1 Alternative denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4.3 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4.4 Dening topologies via continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Hausdor space 354.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.5 Preregularity versus regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.6 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.7 Algebra of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.8 Academic humour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Homeomorphism 395.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2.1 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.5 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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5.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 If and only if 436.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.2.3 Origin of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6.3 Distinction from if and only if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.4 More general usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7 Locally compact space 467.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7.2.1 Compact Hausdor spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.2.2 Locally compact Hausdor spaces that are not compact . . . . . . . . . . . . . . . . . . . 477.2.3 Hausdor spaces that are not locally compact . . . . . . . . . . . . . . . . . . . . . . . . 477.2.4 Non-Hausdor examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

7.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487.3.1 The point at innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487.3.2 Locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

7.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

8 Locally Hausdor space 508.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

9 Locally normal space 519.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519.2 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519.3 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

10 Locally regular space 5310.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5310.2 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5310.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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10.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

11 Metric space 5411.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.3 Examples of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5511.4 Open and closed sets, topology and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.5 Types of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

11.5.1 Complete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.5.2 Bounded and totally bounded spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.5.3 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.5.4 Locally compact and proper spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.5.5 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.5.6 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

11.6 Types of maps between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.6.1 Continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5911.6.2 Uniformly continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5911.6.3 Lipschitz-continuous maps and contractions . . . . . . . . . . . . . . . . . . . . . . . . . 5911.6.4 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6011.6.5 Quasi-isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

11.7 Notions of metric space equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6011.8 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6011.9 Distance between points and sets; Hausdor distance and Gromov metric . . . . . . . . . . . . . . 6111.10Product metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

11.10.1 Continuity of distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6111.11Quotient metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6211.12Generalizations of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

11.12.1 Metric spaces as enriched categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6211.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6311.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6311.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6411.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

12 Metrization theorem 6512.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6512.2 Metrization theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6512.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.4 Examples of non-metrizable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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13.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6713.2 Examples of normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6813.3 Examples of non-normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6813.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.5 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.6 Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

14 Separated sets 7014.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7014.2 Relation to separation axioms and separated spaces . . . . . . . . . . . . . . . . . . . . . . . . . 7114.3 Relation to connected spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7114.4 Relation to topologically distinguishable points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7114.5 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

15 Separation axiom 7215.1 Preliminary denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7215.2 Main denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7315.3 Relationships between the axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7415.4 Other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7415.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7515.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7515.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

16 Subspace topology 7816.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7816.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7816.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7916.4 Preservation of topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8016.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8016.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

17 T1 space 8117.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8117.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8117.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8217.4 Generalisations to other kinds of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8317.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

18 Topological space 8418.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

18.1.1 Neighbourhoods denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8418.1.2 Open sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

vi CONTENTS

18.1.3 Closed sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8618.1.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

18.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8618.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8618.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8718.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8818.6 Classication of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8818.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8818.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8818.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8818.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8918.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8918.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8918.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

19 Uniform space 9119.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

19.1.1 Entourage denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9119.1.2 Pseudometrics denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9219.1.3 Uniform cover denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

19.2 Topology of uniform spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9219.2.1 Uniformizable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

19.3 Uniform continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9319.4 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

19.4.1 Hausdor completion of a uniform space . . . . . . . . . . . . . . . . . . . . . . . . . . . 9419.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9419.6 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9519.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9519.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

20 Vacuous truth 9620.1 Scope of the concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9720.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9720.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9720.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9720.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 98

20.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9820.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10020.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Chapter 1

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

1

2 CHAPTER 1. COMPACT SPACE

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.

1.1 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.

1.2. BASIC EXAMPLES 3

1.2 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.

1.3 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.

1.3.1 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.


1.3.2 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.

1.4. PROPERTIES OF COMPACT SPACES 5

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.

1.3.3 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:

1.4 Properties of compact spaces

1.4.1 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.

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


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.

1.4.3 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]

1.5 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)

1.6. SEE ALSO 7

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.

1.5.1 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.

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


1.7 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)

1.8 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.

1.9. EXTERNAL LINKS 9

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

1.9 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 2

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.

10

2.1. FORMAL DEFINITION 11

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.

2.1 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.

2.1.1 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.

2.1.2 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.

2.2 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].

12 CHAPTER 2. CONNECTED SPACE

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).

2.3 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.

2.4. ARC CONNECTEDNESS 13

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

2.4 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'


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]

2.6. SET OPERATIONS 15

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 disjoint open sets, e.g. YX1 = Z1Z2. Because Y is connected, it must be


entirely contained in one of these components, say Z1, and thus Z2 is contained in X1. Now we know that:

X = (YX1)X2 = (Z1Z2)X2 = (Z1X2)(Z2X1)

The two sets in the last union are disjoint and open in X, so there is a separation of X, contradicting the fact that X isconnected.

2.7 TheoremsMain theorem of connectedness redirects to here.

Main theorem: Let X and Y be topological spaces and let f : X Y be a continuous function. If X is (path-)connected then the image f(X) is (path-)connected. This result can be considered a generalization of theintermediate value theorem.

Every path-connected space is connected.

Every locally path-connected space is locally connected.

A locally path-connected space is path-connected if and only if it is connected.

The closure of a connected subset is connected.

The connected components are always closed (but in general not open)

The connected components of a locally connected space are also open.

The connected components of a space are disjoint unions of the path-connected components (which in generalare neither open nor closed).

Every quotient of a connected (resp. locally connected, path-connected, locally path-connected) space is con-nected (resp. locally connected, path-connected, locally path-connected).

Every product of a family of connected (resp. path-connected) spaces is connected (resp. path-connected).

Every open subset of a locally connected (resp. locally path-connected) space is locally connected (resp. locallypath-connected).

Every manifold is locally path-connected.

2.8 GraphsGraphs have path connected subsets, namely those subsets for which every pair of points has a path of edges joiningthem. But it is not always possible to nd a topology on the set of points which induces the same connected sets. The5-cycle graph (and any n-cycle with n>3 odd) is one such example.As a consequence, a notion of connectedness can be formulated independently of the topology on a space. To wit,there is a category of connective spaces consisting of sets with collections of connected subsets satisfying connectivityaxioms; their morphisms are those functions which map connected sets to connected sets (Muscat & Buhagiar 2006).Topological spaces and graphs are special cases of connective spaces; indeed, the nite connective spaces are preciselythe nite graphs.However, every graph can be canonically made into a topological space, by treating vertices as points and edges ascopies of the unit interval (see topological graph theory#Graphs as topological spaces). Then one can show that thegraph is connected (in the graph theoretical sense) if and only if it is connected as a topological space.

2.9. STRONGER FORMS OF CONNECTEDNESS 17

2.9 Stronger forms of connectednessThere are stronger forms of connectedness for topological spaces, for instance:

If there exist no two disjoint non-empty open sets in a topological space, X, X must be connected, and thushyperconnected spaces are also connected.

Since a simply connected space is, by denition, also required to be path connected, any simply connected spaceis also connected. Note however, that if the path connectedness requirement is dropped from the denitionof simple connectivity, a simply connected space does not need to be connected.

Yet stronger versions of connectivity include the notion of a contractible space. Every contractible space ispath connected and thus also connected.

In general, note that any path connected space must be connected but there exist connected spaces that are not pathconnected. The deleted comb space furnishes such an example, as does the above-mentioned topologists sine curve.

2.10 See also uniformly connected space locally connected space connected component (graph theory) n-connected Connectedness locus Extremally disconnected space

2.11 References

2.11.1 Notes[1] George F. Simmons (1968). Introduction to Topology and Modern Analysis. McGraw Hill Book Company. p. 144. ISBN

0-89874-551-9.

[2] Charles Weibel, The K-book: An introduction to algebraic K-theory

[3] Credit: Saaqib Mahmuud and Henno Brandsma at Math StackExchange.

[4] Credit: Marek at Math StackExchange

2.11.2 General references Munkres, James R. (2000). Topology, Second Edition. Prentice Hall. ISBN 0-13-181629-2. Weisstein, Eric W., Connected Set, MathWorld. V. I. Malykhin (2001), Connected space, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,

ISBN 978-1-55608-010-4 Muscat, J; Buhagiar, D (2006). Connective Spaces (PDF). Mem. Fac. Sci. Eng. Shimane Univ., Series B:Math. Sc. 39: 113..

Chapter 3

Continuous function

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

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

3.2 Real-valued continuous functions

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

f : I ! R:

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

18

3.2. REAL-VALUED CONTINUOUS FUNCTIONS 19

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

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

Denition in terms of limits of functions

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

limx!c f(x) = f(c):

In detail this means three conditions: rst, f has to be dened at c. Second, the limit on the left hand side of thatequation has to exist. Third, the value of this limit must equal f(c).If the point c in the domain of f is not a limit point of the domain, then the above condition is vacuously true, since xcannot approach c through values not equal to c. Thus, for example, function whose domain is the set of all integersis continuous at every point of its domain.The function f is said to be continuous if it is continuous at every point of its domain; otherwise, it is discontinuous.

Denition in terms of limits of sequences

One can instead require that for any sequence (xn)n2N of points in the domain which converges to c, the corre-sponding sequence (f(xn))n2N converges to f(c). In mathematical notation, 8(xn)n2N I : limn!1 xn = c )limn!1 f(xn) = f(c) :

Weierstrass denition (epsilondelta) of continuous functions

Explicitly including the denition of the limit of a function, we obtain a self-contained denition: Given a functionf as above and an element c of the domain I, f is said to be continuous at the point c if the following holds: For anynumber > 0, however small, there exists some number > 0 such that for all x in the domain of f with c < x 0 there exists a > 0 such that for allx I,:

jx cj < ) jf(x) f(c)j < ":More intuitively, we can say that if we want to get all the f(x) values to stay in some small neighborhood around f(c),we simply need to choose a small enough neighborhood for the x values around c, and we can do that no matter howsmall the f(x) neighborhood is; f is then continuous at c.Note. It does not necessarily mean that when x and y are getting closer and closer to each other then so do f(x) andf(y). For instance, if f is the reciprocal function on reals that are not equal zero (so f is a continuous function) and xasserts consecutive values of 1/n and y asserts consecutive values of 1/n, where n diverges to innity, then x and yare are both in the domain of f and are getting closer and closer to each other but the distance between f(x) = 1/x =n and f(y) = 1/y = -n diverges to innity.In modern terms, this is generalized by the denition of continuity of a function with respect to a basis for the topology,here the metric topology.

20 CHAPTER 3. CONTINUOUS FUNCTION

1 2 3 4

1

2

3

4

2

2+

y

x

f(2)+ f(2)

Illustration of the --denition: for =0.5, c=2, the value =0.5 satises the condition of the denition.

Denition using oscillation

Continuity can also be dened in terms of oscillation: a function f is continuous at a point x0 if and only if itsoscillation at that point is zero;[3] in symbols, !f (x0) = 0:A benet of this denition is that it quanties discontinuity:the oscillation gives how much the function is discontinuous at a point.This denition is useful in descriptive set theory to study the set of discontinuities and continuous points the con-tinuous points are the intersection of the sets where the oscillation is less than (hence a G set) and gives a veryquick proof of one direction of the Lebesgue integrability condition.[4]

The oscillation is equivalent to the - denition by a simple re-arrangement, and by using a limit (lim sup, lim inf) todene oscillation: if (at a given point) for a given 0 there is no that satises the - denition, then the oscillationis at least 0, and conversely if for every there is a desired , the oscillation is 0. The oscillation denition can benaturally generalized to maps from a topological space to a metric space.

Denition using the hyperreals

Cauchy dened continuity of a function in the following intuitive terms: an innitesimal change in the independent


f(a)

f(b)

0 p

The failure of a function to be continuous at a point is quantied by its oscillation.

variable corresponds to an innitesimal change of the dependent variable (see Cours d'analyse, page 34). Non-standard analysis is a way of making this mathematically rigorous. The real line is augmented by the addition ofinnite and innitesimal numbers to form the hyperreal numbers. In nonstandard analysis, continuity can be denedas follows.

A real-valued function f is continuous at x if its natural extension to the hyperreals has the property thatfor all innitesimal dx, f(x+dx) f(x) is innitesimal[5]

(see microcontinuity). In other words, an innitesimal increment of the independent variable always produces to aninnitesimal change of the dependent variable, giving a modern expression to Augustin-Louis Cauchy's denition ofcontinuity.

3.2.2 Examples

All polynomial functions, such as f(x) = x3 + x2 - 5x + 3 (pictured), are continuous. This is a consequence of the factthat, given two continuous functions


y=(x+3)(x1)2

4 3 2 1 0 1 2

10

5

5

10

15

20

25

The graph of a cubic function has no jumps or holes. The function is continuous.

f; g : I ! Rdened on the same domain I, then the sum f + g, and the product fg of the two functions are continuous (on thesame domain I). Moreover, the function

f

g: fx 2 Ijg(x) 6= 0g ! R; x 7! f(x)

g(x)

is continuous. (The points where g(x) is zero are discarded, as they are not in the domain of f/g.) For example, thefunction (pictured)

f(x) =2x 1x+ 2

is dened for all real numbers x 2 and is continuous at every such point. Thus it is a continuous function. The


X

Y

1-2

2

y = (2x-1)/(x+2)

The graph of a continuous rational function. The function is not dened for x=2. The vertical and horizontal lines are asymptotes.

question of continuity at x = 2 does not arise, since x = 2 is not in the domain of f. There is no continuous functionF: R R that agrees with f(x) for all x 2. The sinc function g(x) = (sin x)/x, dened for all x0 is continuous atthese points. Thus it is a continuous function, too. However, unlike the on of the previous example, this one can beextended to a continuous function on all real numbers, namely

G(x) =

( sin(x)x if x 6= 0

1 if x = 0;

since the limit of g(x), when x approaches 0, is 1. Therefore, the point x=0 is called a removable singularity of g.Given two continuous functions

f : I ! J( R); g : J ! R;the composition

g f : I ! R; x 7! g(f(x))


is continuous.

3.2.3 Non-examples

An example of a discontinuous function is the function f dened by f(x) = 1 if x > 0, f(x) = 0 if x 0. Pick forinstance = 12. There is no -neighborhood around x = 0 that will force all the f(x) values to be within of f(0).Intuitively we can think of this type of discontinuity as a sudden jump in function values. Similarly, the signum orsign function

1

1

y

x

Plot of the signum function. The hollow dots indicate that sgn(x) is 1 for all x>0 and 1 for all x:1 if x > 00 if x = 01 if x < 0

is discontinuous at x = 0 but continuous everywhere else. Yet another example: the function

f(x) =

(sin

1x2

if x 6= 0

0 if x = 0

is continuous everywhere apart from x = 0.Thomaes function,


Plot of Thomaes function for the domain 0


Relation to dierentiability and integrability

Every dierentiable function

f : (a; b)! R

is continuous, as can be shown. The converse does not hold: for example, the absolute value function

f(x) = jxj =(x if x 0x if x < 0

is everywhere continuous. However, it is not dierentiable at x = 0 (but is so everywhere else). Weierstrasss functionis also everywhere continuous but nowhere dierentiable.The derivative f (x) of a dierentiable function f(x) need not be continuous. If f (x) is continuous, f(x) is said to becontinuously dierentiable. The set of such functions is denoted C1((a, b)). More generally, the set of functions

f : ! R

(from an open interval (or open subset of R) to the reals) such that f is n times dierentiable and such that then-th derivative of f is continuous is denoted Cn(). See dierentiability class. In the eld of computer graphics,these three levels are sometimes called G0 (continuity of position), G1 (continuity of tangency), and G2 (continuityof curvature).Every continuous function

f : [a; b]! R

is integrable (for example in the sense of the Riemann integral). The converse does not hold, as the (integrable, butdiscontinuous) sign function shows.

Pointwise and uniform limits

Given a sequence

f1; f2; : : : : I ! R

of functions such that the limit

f(x) := limn!1 fn(x)

exists for all x in I, the resulting function f(x) is referred to as the pointwise limit of the sequence of functions (fn)nN.The pointwise limit function need not be continuous, even if all functions fn are continuous, as the animation at theright shows. However, f is continuous when the sequence converges uniformly, by the uniform convergence theorem.This theorem can be used to show that the exponential functions, logarithms, square root function, trigonometricfunctions are continuous.

3.2.5 Directional and semi-continuity A right-continuous function A left-continuous function

3.3. CONTINUOUS FUNCTIONS BETWEEN METRIC SPACES 27

A sequence of continuous functions f(x) whose (pointwise) limit function f(x) is discontinuous. The convergence is not uniform.

Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity(or right and left continuous functions) and semi-continuity. Roughly speaking, a function is right-continuous if nojump occurs when the limit point is approached from the right. More formally, f is said to be right-continuous at thepoint c if the following holds: For any number > 0 however small, there exists some number > 0 such that for allx in the domain with c < x < c + , the value of f(x) will satisfy

jf(x) f(c)j < ":This is the same condition as for continuous functions, except that it is required to hold for x strictly larger than conly. Requiring it instead for all x with c < x < c yields the notion of left-continuous functions. A function iscontinuous if and only if it is both right-continuous and left-continuous.A function f is lower semi-continuous if, roughly, any jumps that might occur only go down, but not up. That is, forany > 0, there exists some number > 0 such that for all x in the domain with |x c| < , the value of f(x) satises

f(x) f(c) :The reverse condition is upper semi-continuity.

3.3 Continuous functions between metric spacesThe concept of continuous real-valued functions can be generalized to functions between metric spaces. A metricspace is a set X equipped with a function (called metric) dX, that can be thought of as a measurement of the distanceof any two elements in X. Formally, the metric is a function

dX : X X ! Rthat satises a number of requirements, notably the triangle inequality. Given two metric spaces (X, dX) and (Y, dY)and a function


f : X ! Y

then f is continuous at the point c in X (with respect to the given metrics) if for any positive real number , thereexists a positive real number such that all x in X satisfying dX(x, c) < will also satisfy dY(f(x), f(c)) < . As inthe case of real functions above, this is equivalent to the condition that for every sequence (xn) in X with limit lim xn= c, we have lim f(xn) = f(c). The latter condition can be weakened as follows: f is continuous at the point c if andonly if for every convergent sequence (xn) in X with limit c, the sequence (f(xn)) is a Cauchy sequence, and c is inthe domain of f.The set of points at which a function between metric spaces is continuous is a G set this follows from the -denition of continuity.This notion of continuity is applied, for example, in functional analysis. A key statement in this area says that a linearoperator

T : V !W

between normed vector spaces V and W (which are vector spaces equipped with a compatible norm, denoted ||x||) iscontinuous if and only if it is bounded, that is, there is a constant K such that

kT (x)k Kkxk

for all x in V.

3.3.1 Uniform, Hlder and Lipschitz continuityThe concept of continuity for functions between metric spaces can be strengthened in various ways by limiting theway depends on and c in the denition above. Intuitively, a function f as above is uniformly continuous if the does not depend on the point c. More precisely, it is required that for every real number > 0 there exists > 0 suchthat for every c, b X with dX(b, c) < , we have that dY(f(b), f(c)) < . Thus, any uniformly continuous functionis continuous. The converse does not hold in general, but holds when the domain space X is compact. Uniformlycontinuous maps can be dened in the more general situation of uniform spaces.[6]

A function is Hlder continuous with exponent (a real number) if there is a constant K such that for all b and c inX, the inequality

dY (f(b); f(c)) K (dX(b; c))

holds. Any Hlder continuous function is uniformly continuous. The particular case = 1 is referred to as Lipschitzcontinuity. That is, a function is Lipschitz continuous if there is a constant K such that the inequality

dY (f(b); f(c)) K dX(b; c)

holds any b, c in X.[7] The Lipschitz condition occurs, for example, in the PicardLindelf theorem concerning thesolutions of ordinary dierential equations.

3.4 Continuous functions between topological spacesAnother, more abstract, notion of continuity is continuity of functions between topological spaces in which theregenerally is no formal notion of distance, as there is in the case of metric spaces. A topological space is a set Xtogether with a topology on X, which is a set of subsets of X satisfying a few requirements with respect to their unionsand intersections that generalize the properties of the open balls in metric spaces while still allowing to talk about

3.4. CONTINUOUS FUNCTIONS BETWEEN TOPOLOGICAL SPACES 29

For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph, so thatthe graph always remains entirely outside the cone.

x f(x)

U

X Y

V

f f(U)

Continuity of a function at a point.


the neighbourhoods of a given point. The elements of a topology are called open subsets of X (with respect to thetopology).A function

f : X ! Ybetween two topological spaces X and Y is continuous if for every open set V Y, the inverse image

f1(V ) = fx 2 X j f(x) 2 V gis an open subset of X. That is, f is a function between the sets X and Y (not on the elements of the topology TX),but the continuity of f depends on the topologies used on X and Y.This is equivalent to the condition that the preimages of the closed sets (which are the complements of the opensubsets) in Y are closed in X.An extreme example: if a set X is given the discrete topology (in which every subset is open), all functions

f : X ! Tto any topological space T are continuous. On the other hand, if X is equipped with the indiscrete topology (in whichthe only open subsets are the empty set and X) and the space T set is at least T0, then the only continuous functionsare the constant functions. Conversely, any function whose range is indiscrete is continuous.

3.4.1 Alternative denitionsSeveral equivalent denitions for a topological structure exist and thus there are several equivalent ways to dene acontinuous function.

Neighborhood denition

Neighborhoods continuity for functions between topological spaces (X; TX) and (Y; TY ) at a point may be dened:A function f : X ! Y is continuous at a point x 2 X i for any neighborhood of its image f(x) 2 Y the preimageis again a neighborhood of that point: 8N 2 Nf(x) : f1(N) 2MxAccording to the property that neighborhood systems being upper sets this can be restated as follows:8N 2 Nf(x)9M 2Mx : M f1(N)8N 2 Nf(x)9M 2Mx : f(M) NThe second one being a restatement involving the image rather than the preimage.Literally, this means no matter how small the neighborhood is chosen one can always nd a neighborhood mappedinto it.Besides, theres a simplication involving only open neighborhoods. In fact, they're equivalent:8V 2 TY ; f(x) 2 V 9U 2 TX ; x 2 U : U f1(V )8V 2 TY ; f(x) 2 V 9U 2 TX ; x 2 U : f(U) VThe second one again being a restatement using images rather than preimages.If X and Y are metric spaces, it is equivalent to consider the neighborhood system of open balls centered at x and f(x)instead of all neighborhoods. This gives back the above - denition of continuity in the context of metric spaces.However, in general topological spaces, there is no notion of nearness or distance.Note, however, that if the target space is Hausdor, it is still true that f is continuous at a if and only if the limit off as x approaches a is f(a). At an isolated point, every function is continuous.

Sequences and nets

In several contexts, the topology of a space is conveniently specied in terms of limit points. In many instances, thisis accomplished by specifying when a point is the limit of a sequence, but for some spaces that are too large in some

3.4. CONTINUOUS FUNCTIONS BETWEEN TOPOLOGICAL SPACES 31

sense, one species also when a point is the limit of more general sets of points indexed by a directed set, known asnets. A function is (Heine-)continuous only if it takes limits of sequences to limits of sequences. In the former case,preservation of limits is also sucient; in the latter, a function may preserve all limits of sequences yet still fail to becontinuous, and preservation of nets is a necessary and sucient condition.In detail, a function f: X Y is sequentially continuous if whenever a sequence (xn) in X converges to a limit x,the sequence (f(xn)) converges to f(x). Thus sequentially continuous functions preserve sequential limits. Everycontinuous function is sequentially continuous. If X is a rst-countable space and countable choice holds, then theconverse also holds: any function preserving sequential limits is continuous. In particular, if X is a metric space,sequential continuity and continuity are equivalent. For non rst-countable spaces, sequential continuity might bestrictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.)This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functionspreserve limits of nets, and in fact this property characterizes continuous functions.

Closure operator denition

Instead of specifying the open subsets of a topological space, the topology can also be determined by a closure operator(denoted cl) which assigns to any subset A X its closure, or an interior operator (denoted int), which assigns to anysubset A of X its interior. In these terms, a function

f : (X; cl)! (X 0; cl0)between topological spaces is continuous in the sense above if and only if for all subsets A of X

f(cl(A)) cl0(f(A)):That is to say, given any element x of X that is in the closure of any subset A, f(x) belongs to the closure of f(A).This is equivalent to the requirement that for all subsets A' of X'

f1(cl0(A0)) cl(f1(A0)):Moreover,

f : (X; int)! (X 0; int0)is continuous if and only if

f1(int0(A0)) int(f1(A0))for any subset A' of Y.

3.4.2 PropertiesIf f: X Y and g: Y Z are continuous, then so is the composition g f: X Z. If f: X Y is continuous and

X is compact, then f(X) is compact. X is connected, then f(X) is connected. X is path-connected, then f(X) is path-connected. X is Lindelf, then f(X) is Lindelf. X is separable, then f(X) is separable.


The possible topologies on a xed set X are partially ordered: a topology 1 is said to be coarser than another topology2 (notation: 1 2) if every open subset with respect to 1 is also open with respect to 2. Then, the identity map

idX: (X, 2) (X, 1)

is continuous if and only if 1 2 (see also comparison of topologies). More generally, a continuous function

(X; X)! (Y; Y )stays continuous if the topology Y is replaced by a coarser topology and/or X is replaced by a ner topology.

3.4.3 Ho

locally normal space 1

Documents

properties of compact

hausdor examples

compact groups

compact hausdor spaces477

continuous functions

basic examples

metric spaces

topological spaces