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

2 Cardinal number 32.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Cardinal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4.1 Successor cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.2 Cardinal addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.3 Cardinal multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4.4 Cardinal exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 The continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Continuous function 113.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Real-valued continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.3 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.5 Directional and semi-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Continuous functions between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.1 Uniform, Hlder and Lipschitz continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Continuous functions between topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4.1 Alternative denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

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3.4.3 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4.4 Dening topologies via continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Countable set 284.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.4 Formal denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.5 Minimal model of set theory is countable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.6 Total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Countably compact space 365.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6 Cover (topology) 376.1 Cover in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.2 Renement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.4 Covering dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7 Discrete space 407.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.4 Indiscrete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.5 Quotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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8 Ernst Leonard Lindelf 438.1 Selected bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9 Filter (mathematics) 459.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.2 General denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.3 Filter on a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

9.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.3.2 Filters in model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.3.3 Filters in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

9.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

10 Lawson topology 5110.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

11 Lexicographic order topology on the unit square 5211.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5211.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5211.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5211.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

12 Limit (mathematics) 5312.1 Limit of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5312.2 Limit of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5412.3 Limit as standard part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5512.4 Convergence and xed point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5512.5 Topological net . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5512.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5612.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5612.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

13 Limit point 5713.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5713.2 Types of limit points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5713.3 Some facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5813.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

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13.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

14 Lindelf space 6014.1 Properties of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6014.2 Properties of strongly Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6014.3 Product of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6014.4 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6114.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6114.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6114.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

15 Lindelfs lemma 6215.1 Statement of the lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6215.2 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6215.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

16 List of examples in general topology 6316.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

17 Local homeomorphism 6517.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6517.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6517.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6517.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6617.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

18 Local property 6718.1 Properties of a single space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

18.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6718.2 Properties of a pair of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6718.3 Properties of innite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6818.4 Properties of nite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6818.5 Properties of commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

19 Locally compact space 6919.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6919.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

19.2.1 Compact Hausdor spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7019.2.2 Locally compact Hausdor spaces that are not compact . . . . . . . . . . . . . . . . . . . 7019.2.3 Hausdor spaces that are not locally compact . . . . . . . . . . . . . . . . . . . . . . . . 7019.2.4 Non-Hausdor examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

19.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7119.3.1 The point at innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7119.3.2 Locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

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19.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7219.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

20 Locally connected space 7320.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7420.2 Denitions and rst examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

20.2.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7520.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7520.4 Components and path components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

20.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7620.5 Quasicomponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

20.5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7620.6 More on local connectedness versus weak local connectedness . . . . . . . . . . . . . . . . . . . . 7720.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7720.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7720.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7820.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

21 Locally nite collection 7921.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

21.1.1 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7921.1.2 Second countable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

21.2 Closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8021.3 Countably locally nite collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8021.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

22 Locally nite space 8122.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

23 Metric space 8223.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8223.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8223.3 Examples of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8323.4 Open and closed sets, topology and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 8423.5 Types of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

23.5.1 Complete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8423.5.2 Bounded and totally bounded spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8523.5.3 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8623.5.4 Locally compact and proper spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8623.5.5 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8623.5.6 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

23.6 Types of maps between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8623.6.1 Continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

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23.6.2 Uniformly continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8723.6.3 Lipschitz-continuous maps and contractions . . . . . . . . . . . . . . . . . . . . . . . . . 8723.6.4 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8823.6.5 Quasi-isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

23.7 Notions of metric space equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8823.8 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8823.9 Distance between points and sets; Hausdor distance and Gromov metric . . . . . . . . . . . . . . 8923.10Product metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

23.10.1 Continuity of distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8923.11Quotient metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9023.12Generalizations of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

23.12.1 Metric spaces as enriched categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9023.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9123.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9123.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9223.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

24 Neighbourhood (mathematics) 9324.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9424.2 In a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9424.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9624.4 Topology from neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9624.5 Uniform neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9624.6 Deleted neighbourhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9624.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9624.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

25 Neighbourhood system 9825.1 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9825.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9825.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9825.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9925.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

26 Net (mathematics) 10026.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10026.2 Examples of nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10026.3 Limits of nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10126.4 Examples of limits of nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10126.5 Supplementary denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10126.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10126.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

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26.8 Cauchy nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10326.9 Relation to lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10326.10Limit superior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10326.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

27 Paracompact space 10527.1 Paracompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10527.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10527.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10627.4 Paracompact Hausdor Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

27.4.1 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10727.5 Relationship with compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

27.5.1 Comparison of properties with compactness . . . . . . . . . . . . . . . . . . . . . . . . . 10827.6 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

27.6.1 Denition of relevant terms for the variations . . . . . . . . . . . . . . . . . . . . . . . . . 10927.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10927.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10927.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11027.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

28 Polish space 11128.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11128.2 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11228.3 Polish metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11228.4 Generalizations of Polish spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

28.4.1 Lusin spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11228.4.2 Suslin spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11228.4.3 Radon spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11328.4.4 Polish groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

28.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11328.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

29 Radon measure 11429.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11429.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11429.3 Radon measures on locally compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

29.3.1 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11529.3.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

29.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11629.5 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

29.5.1 Moderated Radon measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11729.5.2 Radon spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

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29.5.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11729.5.4 Metric space structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

29.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11829.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

30 Second-countable space 11930.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

30.1.1 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11930.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12030.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

31 Separable space 12131.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12131.2 Separability versus second countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12131.3 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12231.4 Constructive mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12231.5 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

31.5.1 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12231.5.2 Non-separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

31.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12331.6.1 Embedding separable metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

31.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

32 Subspace topology 12532.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12532.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12532.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12632.4 Preservation of topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12732.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12732.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

33 Topological space 12833.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

33.1.1 Neighbourhoods denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12833.1.2 Open sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12933.1.3 Closed sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13033.1.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

33.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13033.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13033.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13133.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13233.6 Classication of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13233.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

CONTENTS ix

33.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13233.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13233.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13333.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13333.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13333.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

34 Uncountable set 13534.1 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13534.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13534.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13534.4 Without the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13634.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13634.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13634.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

35 -compact space 13735.1 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13735.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13735.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13835.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13835.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 139

35.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13935.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14335.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Chapter 1

Axiom of countability

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

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

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

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

second-countable space: the topology has a countable base

separable space: there exists a countable dense subspace

Lindelf space: every open cover has a countable subcover

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

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

Every rst countable space is sequential.

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

Every -compact space is Lindelf.

Every metric space is rst countable.

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

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

1

2 CHAPTER 1. AXIOM OF COUNTABILITY

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

9780080933795.

Chapter 2

Cardinal number

This article is about the mathematical concept. For number words indicating quantity (three apples, four birds,etc.), see Cardinal number (linguistics).

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used tomeasure the cardinality (size) of sets. The cardinality of a nite set is a natural number: the number of elements inthe set. The transnite cardinal numbers describe the sizes of innite sets.Cardinality is dened in terms of bijective functions. Two sets have the same cardinality if, and only if, there is aone-to-one correspondence (bijection) between the elements of the two sets. In the case of nite sets, this agrees withthe intuitive notion of size. In the case of innite sets, the behavior is more complex. A fundamental theorem due toGeorg Cantor shows that it is possible for innite sets to have dierent cardinalities, and in particular the cardinalityof the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a propersubset of an innite set to have the same cardinality as the original set, something that cannot happen with propersubsets of nite sets.There is a transnite sequence of cardinal numbers:

0; 1; 2; 3; : : : ; n; : : : ;@0;@1;@2; : : : ;@; : : : :This sequence starts with the natural numbers including zero (nite cardinals), which are followed by the alephnumbers (innite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal numbers. Under theassumption of the axiom of choice, this transnite sequence includes every cardinal number. If one rejects thataxiom, the situation is more complicated, with additional innite cardinals that are not alephs.Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics includingcombinatorics, abstract algebra, and mathematical analysis. In category theory, the cardinal numbers form a skeletonof the category of sets.

2.1 HistoryThe notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 18741884. Cardinality can be used to compare an aspect of nite sets; e.g. the sets {1,2,3} and {4,5,6} are not equal,but have the same cardinality, namely three (this is established by the existence of a bijection, i.e. a one-to-onecorrespondence, between the two sets; e.g. {1->4, 2->5, 3->6}).Cantor applied his concept of bijection to innite sets;[1] e.g. the set of natural numbers N = {0, 1, 2, 3, ...}. Thus,all sets having a bijection with N he called denumerable (countably innite) sets and they all have the same cardinalnumber. This cardinal number is called@0 , aleph-null. He called the cardinal numbers of these innite sets, transnitecardinal numbers.Cantor proved that any unbounded subset of N has the same cardinality as N, even though this might appear to runcontrary to intuition. He also proved that the set of all ordered pairs of natural numbers is denumerable (whichimplies that the set of all rational numbers is denumerable), and later proved that the set of all algebraic numbers isalso denumerable. Each algebraic number z may be encoded as a nite sequence of integers which are the coecients

3

4 CHAPTER 2. CARDINAL NUMBER

X 1

2

3

4

YD

B

C

A

A bijective function, f: X Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to thecardinal number 4.

in the polynomial equation of which it is the solution, i.e. the ordered n-tuple (a0, a1, ..., an), ai Z together with apair of rationals (b0, b1) such that z is the unique root of the polynomial with coecients (a0, a1, ..., an) that lies inthe interval (b0, b1).In his 1874 paper, Cantor proved that there exist higher-order cardinal numbers by showing that the set of real numbershas cardinality greater than that of N. His original presentation used a complex argument with nested intervals, but inan 1891 paper he proved the same result using his ingenious but simple diagonal argument. The new cardinal numberof the set of real numbers is called the cardinality of the continuum and Cantor used the symbol c for it.Cantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallesttransnite cardinal number ( @0 , aleph-null) and that for every cardinal number, there is a next-larger cardinal

(@1;@2;@3; ):

His continuum hypothesis is the proposition that c is the same as @1 . This hypothesis has been found to be inde-pendent of the standard axioms of mathematical set theory; it can neither be proved nor disproved from the standardassumptions.

2.2. MOTIVATION 5

Aleph null, the smallest innite cardinal

2.2 Motivation

In informal use, a cardinal number is what is normally referred to as a counting number, provided that 0 is included:0, 1, 2, .... They may be identied with the natural numbers beginning with 0. The counting numbers are exactlywhat can be dened formally as the nite cardinal numbers. Innite cardinals only occur in higher-level mathematicsand logic.More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe theposition of an element in a sequence. For nite sets and sequences it is easy to see that these two notions coincide,since for every number describing a position in a sequence we can construct a set which has exactly the right size,e.g. 3 describes the position of 'c' in the sequence , and we can construct the set {a,b,c} which has3 elements. However when dealing with innite sets it is essential to distinguish between the two the two notionsare in fact dierent for innite sets. Considering the position aspect leads to ordinal numbers, while the size aspect isgeneralized by the cardinal numbers described here.The intuition behind the formal denition of cardinal is the construction of a notion of the relative size or bigness ofa set without reference to the kind of members which it has. For nite sets this is easy; one simply counts the numberof elements a set has. In order to compare the sizes of larger sets, it is necessary to appeal to more subtle notions.


A set Y is at least as big as a set X if there is an injective mapping from the elements of X to the elements of Y.An injective mapping identies each element of the set X with a unique element of the set Y. This is most easilyunderstood by an example; suppose we have the sets X = {1,2,3} and Y = {a,b,c,d}, then using this notion of size wewould observe that there is a mapping:

1 a2 b3 c

which is injective, and hence conclude that Y has cardinality greater than or equal to X. Note the element d has noelement mapping to it, but this is permitted as we only require an injective mapping, and not necessarily an injectiveand onto mapping. The advantage of this notion is that it can be extended to innite sets.We can then extend this to an equality-style relation. Two sets X and Y are said to have the same cardinality if thereexists a bijection between X and Y. By the SchroederBernstein theorem, this is equivalent to there being both aninjective mapping from X to Y and an injective mapping from Y to X. We then write |X| = |Y |. The cardinal numberof X itself is often dened as the least ordinal a with |a| = |X|. This is called the von Neumann cardinal assignment; forthis denition to make sense, it must be proved that every set has the same cardinality as some ordinal; this statement isthe well-ordering principle. It is however possible to discuss the relative cardinality of sets without explicitly assigningnames to objects.The classic example used is that of the innite hotel paradox, also called Hilberts paradox of the Grand Hotel.Suppose you are an innkeeper at a hotel with an innite number of rooms. The hotel is full, and then a new guestarrives. It is possible to t the extra guest in by asking the guest who was in room 1 to move to room 2, the guest inroom 2 to move to room 3, and so on, leaving room 1 vacant. We can explicitly write a segment of this mapping:

1 22 33 4...n n + 1...

In this way we can see that the set {1,2,3,...} has the same cardinality as the set {2,3,4,...} since a bijection betweenthe rst and the second has been shown. This motivates the denition of an innite set being any set which has aproper subset of the same cardinality; in this case {2,3,4,...} is a proper subset of {1,2,3,...}.When considering these large objects, we might also want to see if the notion of counting order coincides with thatof cardinal dened above for these innite sets. It happens that it doesn't; by considering the above example we cansee that if some object one greater than innity exists, then it must have the same cardinality as the innite setwe started out with. It is possible to use a dierent formal notion for number, called ordinals, based on the ideasof counting and considering each number in turn, and we discover that the notions of cardinality and ordinality aredivergent once we move out of the nite numbers.It can be proved that the cardinality of the real numbers is greater than that of the natural numbers just described.This can be visualized using Cantors diagonal argument; classic questions of cardinality (for instance the continuumhypothesis) are concerned with discovering whether there is some cardinal between some pair of other innite car-dinals. In more recent times mathematicians have been describing the properties of larger and larger cardinals.Since cardinality is such a common concept in mathematics, a variety of names are in use. Sameness of cardinality issometimes referred to as equipotence, equipollence, or equinumerosity. It is thus said that two sets with the samecardinality are, respectively, equipotent, equipollent, or equinumerous.

2.3 Formal denitionFormally, assuming the axiom of choice, the cardinality of a set X is the least ordinal such that there is a bijectionbetween X and . This denition is known as the von Neumann cardinal assignment. If the axiom of choice is not

2.4. CARDINAL ARITHMETIC 7

assumed we need to do something dierent. The oldest denition of the cardinality of a set X (implicit in Cantor andexplicit in Frege and Principia Mathematica) is as the class [X] of all sets that are equinumerous with X. This doesnot work in ZFC or other related systems of axiomatic set theory because if X is non-empty, this collection is toolarge to be a set. In fact, for X there is an injection from the universe into [X] by mapping a set m to {m} Xand so by the axiom of limitation of size, [X] is a proper class. The denition does work however in type theory andin New Foundations and related systems. However, if we restrict from this class to those equinumerous with X thathave the least rank, then it will work (this is a trick due to Dana Scott:[2] it works because the collection of objectswith any given rank is a set).Formally, the order among cardinal numbers is dened as follows: |X| |Y | means that there exists an injectivefunction from X to Y. The CantorBernsteinSchroeder theorem states that if |X| |Y | and |Y | |X| then |X| = |Y |.The axiom of choice is equivalent to the statement that given two sets X and Y, either |X| |Y | or |Y | |X|.[3][4]

A set X is Dedekind-innite if there exists a proper subset Y of X with |X| = |Y |, and Dedekind-nite if such a subsetdoesn't exist. The nite cardinals are just the natural numbers, i.e., a set X is nite if and only if |X| = |n| = n forsome natural number n. Any other set is innite. Assuming the axiom of choice, it can be proved that the Dedekindnotions correspond to the standard ones. It can also be proved that the cardinal @0 (aleph null or aleph-0, where alephis the rst letter in the Hebrew alphabet, represented @ ) of the set of natural numbers is the smallest innite cardinal,i.e. that any innite set has a subset of cardinality @0: The next larger cardinal is denoted by @1 and so on. For everyordinal there is a cardinal number @; and this list exhausts all innite cardinal numbers.

2.4 Cardinal arithmeticWe can dene arithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers.It can be shown that for nite cardinals these operations coincide with the usual operations for natural numbers.Furthermore, these operations share many properties with ordinary arithmetic.

2.4.1 Successor cardinalFor more details on this topic, see Successor cardinal.

If the axiom of choice holds, every cardinal has a successor + > , and there are no cardinals between and itssuccessor. For nite cardinals, the successor is simply + 1. For innite cardinals, the successor cardinal diersfrom the successor ordinal.

2.4.2 Cardinal additionIf X and Y are disjoint, addition is given by the union of X and Y. If the two sets are not already disjoint, then theycan be replaced by disjoint sets of the same cardinality, e.g., replace X by X{0} and Y by Y{1}.

jXj+ jY j = jX [ Y j:Zero is an additive identity + 0 = 0 + = .Addition is associative ( + ) + = + ( + ).Addition is commutative + = + .Addition is non-decreasing in both arguments:

( )! ((+ + ) and ( + + )):Assuming the axiom of choice, addition of innite cardinal numbers is easy. If either or is innite, then

+ = maxf; g :


Subtraction

Assuming the axiom of choice and, given an innite cardinal and a cardinal , there exists a cardinal such that + = if and only if . It will be unique (and equal to ) if and only if < .

2.4.3 Cardinal multiplicationThe product of cardinals comes from the cartesian product.

jXj jY j = jX Y j0 = 0 = 0. = 0 ( = 0 or = 0).One is a multiplicative identity 1 = 1 = .Multiplication is associative () = ().Multiplication is commutative = .Multiplication is non-decreasing in both arguments: ( and ).Multiplication distributes over addition: ( + ) = + and ( + ) = + .Assuming the axiom of choice, multiplication of innite cardinal numbers is also easy. If either or is innite andboth are non-zero, then

= maxf; g:

Division

Assuming the axiom of choice and, given an innite cardinal and a non-zero cardinal , there exists a cardinal such that = if and only if . It will be unique (and equal to ) if and only if < .

2.4.4 Cardinal exponentiationExponentiation is given by

jXjjY j = XY where XY is the set of all functions from Y to X.

0 = 1 (in particular 00 = 1), see empty function.If 1 , then 0 = 0.1 = 1.1 = . + = . = ().() = .

Exponentiation is non-decreasing in both arguments:

(1 and ) ( ) and( ) ( ).

2.5. THE CONTINUUM HYPOTHESIS 9

Note that 2|X| is the cardinality of the power set of the set X and Cantors diagonal argument shows that 2|X| > |X| forany set X. This proves that no largest cardinal exists (because for any cardinal , we can always nd a larger cardinal2). In fact, the class of cardinals is a proper class. (This proof fails in some set theories, notably New Foundations.)All the remaining propositions in this section assume the axiom of choice:

If and are both nite and greater than 1, and is innite, then = .If is innite and is nite and non-zero, then = .

If 2 and 1 and at least one of them is innite, then:

Max (, 2) Max (2, 2).

Using Knigs theorem, one can prove < cf() and < cf(2) for any innite cardinal , where cf() is the conalityof .

Roots

Assuming the axiom of choice and, given an innite cardinal and a nite cardinal greater than 0, the cardinal satisfying = will be .

Logarithms

Assuming the axiom of choice and, given an innite cardinal and a nite cardinal greater than 1, there may ormay not be a cardinal satisfying = . However, if such a cardinal exists, it is innite and less than , and anynite cardinality greater than 1 will also satisfy = .The logarithm of an innite cardinal number is dened as the least cardinal number such that 2. Logarithmsof innite cardinals are useful in some elds of mathematics, for example in the study of cardinal invariants oftopological spaces, though they lack some of the properties that logarithms of positive real numbers possess.[5][6][7]

2.5 The continuum hypothesisThe continuum hypothesis (CH) states that there are no cardinals strictly between @0 and 2@0 : The latter cardinalnumber is also often denoted by c ; it is the cardinality of the continuum (the set of real numbers). In this case2@0 = @1: The generalized continuum hypothesis (GCH) states that for every innite set X, there are no cardinalsstrictly between | X | and 2| X |. The continuum hypothesis is independent of the usual axioms of set theory, theZermelo-Fraenkel axioms together with the axiom of choice (ZFC).

2.6 See also

2.7 ReferencesNotes

[1] Dauben 1990, pg. 54

[2] Deiser, Oliver (May 2010). On the Development of the Notion of a Cardinal Number. History and Philosophy of Logic31 (2): 123143. doi:10.1080/01445340903545904.

[3] Enderton, Herbert. Elements of Set Theory, Academic Press Inc., 1977. ISBN 0-12-238440-7

[4] Friedrich M. Hartogs (1915), Felix Klein, Walther von Dyck, David Hilbert, Otto Blumenthal, ed., "ber das Problem derWohlordnung, Math. Ann (Leipzig: B. G. Teubner), Bd. 76 (4): 438443, ISSN 0025-5831


[5] Robert A. McCoy and Ibula Ntantu, Topological Properties of Spaces of Continuous Functions, Lecture Notes in Mathe-matics 1315, Springer-Verlag.

[6] Eduard ech, Topological Spaces, revised by Zdenek Frolk and Miroslav Katetov, John Wiley & Sons, 1966.

[7] D.A. Vladimirov, Boolean Algebras in Analysis, Mathematics and Its Applications, Kluwer Academic Publishers.

Bibliography

Dauben, Joseph Warren (1990), Georg Cantor: His Mathematics and Philosophy of the Innite, Princeton:Princeton University Press, ISBN 0691-02447-2

Hahn, Hans, Innity, Part IX, Chapter 2, Volume 3 of The World of Mathematics. New York: Simon andSchuster, 1956.

Halmos, Paul, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).

2.8 External links Hazewinkel, Michiel, ed. (2001), Cardinal number, Encyclopedia of Mathematics, Springer, ISBN 978-1-

55608-010-4 Weisstein, Eric W., Cardinal Number, MathWorld. Cardinality at ProvenMath proofs of the basic theorems on cardinality.

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

11

12 CHAPTER 3. CONTINUOUS FUNCTION

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.

3.2. REAL-VALUED CONTINUOUS FUNCTIONS 13

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) =

(sin1x2

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


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

3.4. CONTINUOUS FUNCTIONS BETWEEN TOPOLOGICAL SPACES 21

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


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.

3.4. CONTINUOUS FUNCTIONS BETWEEN TOPOLOGICAL SPACES 23

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


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.

3.5. RELATED NOTIONS 25

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 HomeomorphismsSymmetric to the concept of a continuous map is an open map, for which images of open sets are open. In fact, if anopen map f has an inverse function, that inverse is continuous, and if a continuous map g has an inverse, that inverseis open. Given a bijective function f between two topological spaces, the inverse function f1 need not be continuous.A bijective continuous function with continuous inverse function is called a homeomorphism.If a continuous bijection has as its domain a compact space and its codomain is Hausdor, then it is a homeomorphism.

3.4.4 Dening topologies via continuous functionsGiven a function

f : X ! S;where X is a topological space and S is a set (without a specied topology), the nal topology on S is dened by lettingthe open sets of S be those subsets A of S for which f1(A) is open in X. If S has an existing topology, f is continuouswith respect to this topology if and only if the existing topology is coarser than the nal topology on S. Thus the naltopology can be characterized as the nest topology on S that makes f continuous. If f is surjective, this topology iscanonically identied with the quotient topology under the equivalence relation dened by f.Dually, for a function f from a set S to a topological space, the initial topology on S has as open subsets A of S thosesubsets for which f(A) is open in X. If S has an existing topology, f is continuous with respect to this topology if andonly if the existing topology is ner than the initial topology on S. Thus the initial topology can be characterized asthe coarsest topology on S that makes f continuous. If f is injective, this topology is canonically identied with thesubspace topology of S, viewed as a subset of X.More generally, given a set S, specifying the set of continuous functions

S ! Xinto all topological spaces X denes a topology. Dually, a similar idea can be applied to maps

X ! S:This is an instance of a universal property.

3.5 Related notionsVarious other mathematical domains use the concept of continuity in dierent, but related meanings. For example,in order theory, an order-preserving function f: X Y between two complete lattices X and Y (particular typesof partially ordered sets) is continuous if for each subset A of X, we have sup(f(A)) = f(sup(A)). Here sup is the


supremum with respect to the orderings in X and Y, respectively. Applying this to the complete lattice consisting ofthe open subsets of a topological space, this gives back the notion of continuity for maps between topological spaces.In category theory, a functor

F : C ! Dbetween two categories is called continuous, if it commutes with small limits. That is to say,

lim i2I

F (Ci) = F (lim i2I

Ci)

for any small (i.e., indexed by a set I, as opposed to a class) diagram of objects in C .A continuity space is a generalization of metric spaces and posets,[8][9] which uses the concept of quantales, and thatcan be used to unify the notions of metric spaces and domains.[10]

3.6 See also Absolute continuity Classication of discontinuities Coarse function Continuous stochastic process Dini continuity Discrete function Equicontinuity Normal function Piecewise Symmetrically continuous function

3.7 Notes[1] Rusnock, P.; Kerr-Lawson, A. (2005), Bolzano and uniform continuity,HistoriaMathematica 32 (3): 303311, doi:10.1016/j.hm.2004.11.003[2] Lang, Serge (1997), Undergraduate analysis, Undergraduate Texts in Mathematics (2nd ed.), Berlin, New York: Springer-

Verlag, ISBN 978-0-387-94841-6, section II.4[3] Introduction to Real Analysis, updated April 2010, William F. Trench, Theorem 3.5.2, p. 172[4] Introduction to Real Analysis, updated April 2010, William F. Trench, 3.5 A More Advanced Look at the Existence of

the Proper Riemann Integral, pp. 171177[5] http://www.math.wisc.edu/~{}keisler/calc.html[6] Gaal, Steven A. (2009), Point set topology, New York: Dover Publications, ISBN 978-0-486-47222-5, section IV.10[7] Searcid, Mchel (2006), Metric spaces, Springer undergraduate mathematics series, Berlin, New York: Springer-

Verlag, ISBN 978-1-84628-369-7, section 9.4[8] Flagg, R. C. (1997). Quantales and continuity spaces. Algebra Universalis. CiteSeerX: 10 .1 .1 .48 .851.[9] Kopperman, R. (1988). All topologies come from generalized metrics. American Mathematical Monthly 95 (2): 8997.

doi:10.2307/2323060.[10] Flagg, B.; Kopperman, R. (1997). Continuity spaces: Reconciling domains and metric spaces. Theoretical Computer

Science 177 (1): 111138. doi:10.1016/S0304-3975(97)00236-3.

3.8. REFERENCES 27

3.8 References Hazewinkel, Michiel, ed. (2001), Continuous function, Encyclopedia of Mathematics, Springer, ISBN 978-

1-55608-010-4 Visual Calculus by Lawrence S. Husch, University of Tennessee (2001).

Chapter 4

Countable set

Countable redirects here. For the linguistic concept, see Count noun.Not to be confused with (recursively) enumerable sets.

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the setof natural numbers. A countable set is either a nite set or a countably innite set. Whether nite or innite, theelements of a countable set can always be counted one at a time and, although the counting may never nish, everyelement of the set is associated with a natural number.Some authors use countable set to mean innitely countable alone.[1] To avoid this ambiguity, the term at mostcountable may be used when nite sets are included and countably innite, enumerable, or denumerable[2] oth-erwise.The term countable set was originated by Georg Cantor who contrasted sets which are countable with those which areuncountable (a.k.a. nonenumerable and nondenumerable[3]). Today, countable sets are researched by a branch ofmathematics called discrete mathematics.

4.1 DenitionA set S is called countable if there exists an injective function f from S to the natural numbers N = {0, 1, 2, 3, ...}.[4]

If such an f can be found which is also surjective (and therefore bijective), then S is called countably innite.In other words, a set is called countably innite if it has one-to-one correspondence with the natural number set, N.As noted above, this terminology is not universal: Some authors use countable to mean what is here called countablyinnite, and to not include nite sets.For alternative (equivalent) formulations of the denition in terms of a bijective function or a surjective function, seethe section Formal denition and properties below.

4.2 HistoryIn the western world, dierent innities were rst classied by Georg Cantor around 1874.[5]

4.3 IntroductionA set is a collection of elements, and may be described in many ways. One way is simply to list all of its elements;for example, the set consisting of the integers 3, 4, and 5 may be denoted {3, 4, 5}. This is only eective for smallsets, however; for larger sets, this would be time-consuming and error-prone. Instead of listing every single element,sometimes an ellipsis ("...) is used, if the writer believes that the reader can easily guess what is missing; for example,

28

4.4. FORMAL DEFINITION AND PROPERTIES 29

{1, 2, 3, ..., 100} presumably denotes the set of integers from 1 to 100. Even in this case, however, it is still possibleto list all the elements, because the set is nite.Some sets are innite; these sets have more than n elements for any integer n. For example, the set of natural numbers,denotable by {0, 1, 2, 3, 4, 5, ...}, has innitely many elements, and we cannot use any normal number to give itssize. Nonetheless, it turns out that innite sets do have a well-dened notion of size (or more properly, of cardinality,which is the technical term for the number of elements in a set), and not all innite sets have the same cardinality.

YX123

x

246

2x. .

. .

Bijective mapping from integer to even numbers

To understand what this means, we rst examine what it does not mean. For example, there are innitely many oddintegers, innitely many even integers, and (hence) innitely many integers overall. However, it turns out that thenumber of even integers, which is the same as the number of odd integers, is also the same as the number of integersoverall. This is because we arrange things such that for every integer, there is a distinct even integer: ... 24,12, 00, 12, 24, ...; or, more generally, n2n, see picture. What we have done here is arranged the integersand the even integers into a one-to-one correspondence (or bijection), which is a function that maps between two setssuch that each element of each set corresponds to a single element in the other set.However, not all innite sets have the same cardinality. For example, Georg Cantor (who introduced this concept)demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers.A set is countable if: (1) it is nite, or (2) it has the same cardinality (size) as the set of natural numbers. Equivalently, aset is countable if it has the same cardinality as some subset of the set of natural numbers. Otherwise, it is uncountable.

4.4 Formal denition and propertiesBy denition a set S is countable if there exists an injective function f : S N from S to the natural numbers N ={0, 1, 2, 3, ...}.

30 CHAPTER 4. COUNTABLE SET

It might seem natural to divide the sets into dierent classes: put all the sets containing one element together; all thesets containing two elements together; ...; nally, put together all innite sets and consider them as having the samesize. This view is not tenable, however, under the natural denition of size.To elaborate this we need the concept of a bijection. Although a bijection seems a more advanced concept than anumber, the usual development of mathematics in terms of set theory denes functions before numbers, as they arebased on much simpler sets. This is where the concept of a bijection comes in: dene the correspondence

a 1, b 2, c 3

Since every element of {a, b, c} is paired with precisely one element of {1, 2, 3}, and vice versa, this denes abijection.We now generalize this situation and dene two sets to be of the same size if (and only if) there is a bijection betweenthem. For all nite sets this gives us the usual denition of the same size. What does it tell us about the size ofinnite sets?Consider the sets A = {1, 2, 3, ... }, the set of positive integers and B = {2, 4, 6, ... }, the set of even positive integers.We claim that, under our denition, these sets have the same size, and that therefore B is countably innite. Recallthat to prove this we need to exhibit a bijection between them. But this is easy, using n 2n, so that

1 2, 2 4, 3 6, 4 8, ....

As in the earlier example, every element of A has been paired o with precisely one element of B, and vice versa.Hence they have the same size. This gives an example of a set which is of the same size as one of its proper subsets,a situation which is impossible for nite sets.Likewise, the set of all ordered pairs of natural numbers is countably innite, as can be seen by following a path likethe one in the picture:The resulting mapping is like this:

0 (0,0), 1 (1,0), 2 (0,1), 3 (2,0), 4 (1,1), 5 (0,2), 6 (3,0) ....

It is evident that this mapping will cover all such ordered pairs.Interestingly: if you treat each pair as being the numerator and denominator of a vulga

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