Locally finite spaceFrom Wikipedia, the free encyclopedia

Contents

1 a-paracompact space 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Closure (topology) 22.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Point of closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.2 Limit point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.3 Closure of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Closure operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Facts about closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 Categorical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Compact operator 63.1 Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Important properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Origins in integral equation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 Compact operator on Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.5 Completely continuous operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Compact space 104.1 Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.3.1 Open cover definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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4.3.2 Equivalent definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3.3 Compactness of subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.4 Properties of compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4.1 Functions and compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4.2 Compact spaces and set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4.3 Ordered compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.5.1 Algebraic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Compactly embedded 195.1 Definition (topological spaces) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Definition (normed spaces) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6 Cover (topology) 206.1 Cover in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.2 Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.4 Covering dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7 Exhaustion by compact sets 237.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8 Feebly compact space 24

9 Functional analysis 259.1 Normed vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

9.1.1 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.1.2 Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

9.2 Major and foundational results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.2.1 Uniform boundedness principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.2.2 Spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.2.3 Hahn-Banach theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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9.2.4 Open mapping theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.2.5 Closed graph theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.2.6 Other topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

9.3 Foundations of mathematics considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.4 Points of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

10 H-closed space 3110.1 Examples and equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

11 Hemicompact space 3211.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

12 Interior (topology) 3412.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

12.1.1 Interior point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.1.2 Interior of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

12.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.3 Interior operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.4 Exterior of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.5 Interior-disjoint shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

13 k-cell (mathematics) 3913.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.2 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

14 Lebesgue covering dimension 4114.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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14.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.5.1 Historical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.5.2 Modern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

14.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

15 Limit point compact 4315.1 Properties and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4315.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4315.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

16 Lindelöf space 4516.1 Properties of Lindelöf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.2 Properties of strongly Lindelöf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.3 Product of Lindelöf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.4 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

17 Locally compact space 4717.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4717.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

17.2.1 Compact Hausdorff spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4817.2.2 Locally compact Hausdorff spaces that are not compact . . . . . . . . . . . . . . . . . . . 4817.2.3 Hausdorff spaces that are not locally compact . . . . . . . . . . . . . . . . . . . . . . . . 4817.2.4 Non-Hausdorff examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

17.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4917.3.1 The point at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4917.3.2 Locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

17.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5017.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

18 Locally finite 51

19 Locally finite collection 5219.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

19.1.1 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5219.1.2 Second countable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

19.2 Closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5319.3 Countably locally finite collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5319.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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20 Locally finite space 5420.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

21 Mesocompact space 5521.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5521.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

22 Metacompact space 5622.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5622.2 Covering dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5622.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5622.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

23 Orthocompact space 5823.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

24 Paracompact space 5924.1 Paracompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5924.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5924.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6024.4 Paracompact Hausdorff Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

24.4.1 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6124.5 Relationship with compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

24.5.1 Comparison of properties with compactness . . . . . . . . . . . . . . . . . . . . . . . . . 6224.6 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

24.6.1 Definition of relevant terms for the variations . . . . . . . . . . . . . . . . . . . . . . . . . 6324.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6324.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6324.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6424.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

25 Pseudocompact space 6525.1 Properties related to pseudocompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6525.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6525.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

26 Realcompact space 6726.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6726.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6726.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

27 Relatively compact subspace 6927.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6927.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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28 Sequentially compact space 7028.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7028.2 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7028.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7028.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7028.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

29 Strictly singular operator 7229.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

30 Subset 7330.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7430.2 ⊂ and ⊃ symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7430.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7430.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7530.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7530.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7530.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

31 Supercompact space 7731.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7731.2 Some Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7731.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

32 Topological space 7932.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

32.1.1 Neighbourhoods definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7932.1.2 Open sets definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8032.1.3 Closed sets definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8132.1.4 Other definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

32.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8132.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8132.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8232.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8332.6 Classification of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8332.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8332.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8332.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8332.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8432.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8432.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8432.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

CONTENTS vii

33 Topology 8633.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8733.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8833.3 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

33.3.1 Topologies on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9033.3.2 Continuous functions and homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 9133.3.3 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

33.4 Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9133.4.1 General topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9133.4.2 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9233.4.3 Differential topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9233.4.4 Geometric topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9233.4.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

33.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9333.5.1 Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9333.5.2 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9333.5.3 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9333.5.4 Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

33.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9333.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9433.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9533.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

34 Totally bounded space 9634.1 Definition for a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9634.2 Definitions in other contexts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9634.3 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9734.4 Relationships with compactness and completeness . . . . . . . . . . . . . . . . . . . . . . . . . . 9734.5 Use of the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9834.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9834.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9834.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

35 σ-compact space 9935.1 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9935.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9935.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10035.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10035.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 101

35.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10135.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10435.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Chapter 1

a-paracompact space

In mathematics, in the field of topology, a topological space is said to be a-paracompact if every open cover of thespace has a locally finite refinement. In contrast to the definition of paracompactness, the refinement is not requiredto be open.Every paracompact space is a-paracompact, and in regular spaces the two notions coincide.

1.1 References• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

1

Chapter 2

Closure (topology)

For other uses, see Closure (disambiguation).

In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S.The closure of S is also defined as the union of S and its boundary. Intuitively, these are all the points in S and “near”S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to thenotion of interior.

2.1 Definitions

2.1.1 Point of closure

For S a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S(this point may be x itself).This definition generalises to any subset S of a metric space X. Fully expressed, for X a metric space with metric d, xis a point of closure of S if for every r > 0, there is a y in S such that the distance d(x, y) < r. (Again, we may have x= y.) Another way to express this is to say that x is a point of closure of S if the distance d(x, S) := infd(x, s) : s inS = 0.This definition generalises to topological spaces by replacing “open ball” or “ball” with "neighbourhood". Let S bea subset of a topological space X. Then x is a point of closure (or adherent point) of S if every neighbourhood of xcontains a point of S.[1] Note that this definition does not depend upon whether neighbourhoods are required to beopen.

2.1.2 Limit point

The definition of a point of closure is closely related to the definition of a limit point. The difference between thetwo definitions is subtle but important — namely, in the definition of limit point, every neighborhood of the point xin question must contain a point of the set other than x itself.Thus, every limit point is a point of closure, but not every point of closure is a limit point. A point of closure whichis not a limit point is an isolated point. In other words, a point x is an isolated point of S if it is an element of S andif there is a neighbourhood of x which contains no other points of S other than x itself.[2]

For a given set S and point x, x is a point of closure of S if and only if x is an element of S or x is a limit point of S(or both).

2.1.3 Closure of a set

See also: Closure (mathematics)

2

2.2. EXAMPLES 3

The closure of a set S is the set of all points of closure of S, that is, the set S together with all of its limit points.[3]The closure of S is denoted cl(S), Cl(S), S or S− . The closure of a set has the following properties.[4]

• cl(S) is a closed superset of S.

• cl(S) is the intersection of all closed sets containing S.

• cl(S) is the smallest closed set containing S.

• cl(S) is the union of S and its boundary ∂(S).

• A set S is closed if and only if S = cl(S).

• If S is a subset of T, then cl(S) is a subset of cl(T).

• If A is a closed set, then A contains S if and only if A contains cl(S).

Sometimes the second or third property above is taken as the definition of the topological closure, which still makesense when applied to other types of closures (see below).[5]

In a first-countable space (such as a metric space), cl(S) is the set of all limits of all convergent sequences of pointsin S. For a general topological space, this statement remains true if one replaces “sequence” by "net" or "filter".Note that these properties are also satisfied if “closure”, “superset”, “intersection”, “contains/containing”, “smallest”and “closed” are replaced by “interior”, “subset”, “union”, “contained in”, “largest”, and “open”. For more on thismatter, see closure operator below.

2.2 Examples

Consider a sphere in 3 dimensions. Implicitly there are two regions of interest created by this sphere; the sphere itselfand its interior (which is called an open 3-ball). It is useful to be able to distinguish between the interior of 3-ball andthe surface, so we distinguish between the open 3-ball, and the closed 3-ball - the closure of the 3-ball. The closureof the open 3-ball is the open 3-ball plus the surface.In topological space:

• In any space, ∅ = cl(∅) .

• In any space X, X = cl(X).

Giving R and C the standard (metric) topology:

• If X is the Euclidean space R of real numbers, then cl((0, 1)) = [0, 1].

• If X is the Euclidean space R, then the closure of the set Q of rational numbers is the whole space R. We saythat Q is dense in R.

• If X is the complex plane C = R2, then cl(z in C : |z| > 1) = z in C : |z| ≥ 1.

• If S is a finite subset of a Euclidean space, then cl(S) = S. (For a general topological space, this property isequivalent to the T1 axiom.)

On the set of real numbers one can put other topologies rather than the standard one.

• If X = R, where R has the lower limit topology, then cl((0, 1)) = [0, 1).

• If one considers on R the discrete topology in which every set is closed (open), then cl((0, 1)) = (0, 1).

• If one considers on R the trivial topology in which the only closed (open) sets are the empty set and R itself,then cl((0, 1)) = R.

4 CHAPTER 2. CLOSURE (TOPOLOGY)

These examples show that the closure of a set depends upon the topology of the underlying space. The last twoexamples are special cases of the following.

• In any discrete space, since every set is closed (and also open), every set is equal to its closure.• In any indiscrete space X, since the only closed sets are the empty set and X itself, we have that the closureof the empty set is the empty set, and for every non-empty subset A of X, cl(A) = X. In other words, everynon-empty subset of an indiscrete space is dense.

The closure of a set also depends upon in which space we are taking the closure. For example, if X is the set ofrational numbers, with the usual relative topology induced by the Euclidean space R, and if S = q in Q : q2 > 2, q >0, then S is closed in Q, and the closure of S in Q is S; however, the closure of S in the Euclidean space R is the setof all real numbers greater than or equal to

√2.

2.3 Closure operator

See also: Closure operator

A closure operator on a set X is a mapping of the power set of X, P(X) , into itself which satisfies the Kuratowskiclosure axioms.Given a topological space (X, T ) , the mapping − : S → S− for all S ⊆ X is a closure operator on X. Conversely, if cis a closure operator on a set X, a topological space is obtained by defining the sets S with c(S) = S as closed sets (sotheir complements are the open sets of the topology).[6]

The closure operator − is dual to the interior operator o, in the sense that

S− = X \ (X \ S)o

and also

So = X \ (X \ S)−

where X denotes the underlying set of the topological space containing S, and the backslash refers to the set-theoreticdifference.Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be easily translated intothe language of interior operators, by replacing sets with their complements.

2.4 Facts about closures

The set S is closed if and only if Cl(S) = S . In particular:

• The closure of the empty set is the empty set;• The closure of X itself is X .• The closure of an intersection of sets is always a subset of (but need not be equal to) the intersection of theclosures of the sets.

• In a union of finitely many sets, the closure of the union and the union of the closures are equal; the union ofzero sets is the empty set, and so this statement contains the earlier statement about the closure of the emptyset as a special case.

• The closure of the union of infinitely many sets need not equal the union of the closures, but it is always asuperset of the union of the closures.

If A is a subspace ofX containing S , then the closure of S computed in A is equal to the intersection of A and theclosure of S computed inX : ClA(S) = A ∩ClX(S) . In particular, S is dense in A if and only if A is a subset ofClX(S) .

2.5. CATEGORICAL INTERPRETATION 5

2.5 Categorical interpretation

One may elegantly define the closure operator in terms of universal arrows, as follows.The powerset of a set X may be realized as a partial order category P in which the objects are subsets and themorphisms are inclusions A → B whenever A is a subset of B. Furthermore, a topology T on X is a subcategory ofP with inclusion functor I : T → P . The set of closed subsets containing a fixed subset A ⊆ X can be identifiedwith the comma category (A ↓ I) . This category — also a partial order — then has initial object Cl(A). Thus thereis a universal arrow from A to I, given by the inclusion A→ Cl(A) .Similarly, since every closed set containing X \ A corresponds with an open set contained in A we can interpret thecategory (I ↓ X \A) as the set of open subsets contained in A, with terminal object int(A) , the interior of A.All properties of the closure can be derived from this definition and a few properties of the above categories. More-over, this definition makes precise the analogy between the topological closure and other types of closures (forexample algebraic), since all are examples of universal arrows.

2.6 See also• Closure algebra

2.7 Notes[1] Schubert, p. 20

[2] Kuratowski, p. 75

[3] Hocking Young, p. 4

[4] Croom, p. 104

[5] Gemignani, p. 55, Pervin, p. 40 and Baker, p. 38 use the second property as the definition.

[6] Pervin, p. 41

2.8 References• Baker, Crump W. (1991), Introduction to Topology, Wm. C. Brown Publisher, ISBN 0-697-05972-3

• Croom, Fred H. (1989), Principles of Topology, Saunders College Publishing, ISBN 0-03-012813-7

• Gemignani, Michael C. (1990) [1967], Elementary Topology (2nd ed.), Dover, ISBN 0-486-66522-4

• Hocking, John G.; Young, Gail S. (1988) [1961], Topology, Dover, ISBN 0-486-65676-4

• Kuratowski, K. (1966), Topology I, Academic Press

• Pervin, William J. (1965), Foundations of General Topology, Academic Press

• Schubert, Horst (1968), Topology, Allyn and Bacon

2.9 External links• Hazewinkel, Michiel, ed. (2001), “Closure of a set”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 3

Compact operator

In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space Xto another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset ofY. Such an operator is necessarily a bounded operator, and so continuous.Any bounded operator L that has finite rank is a compact operator; indeed, the class of compact operators is a naturalgeneralisation of the class of finite-rank operators in an infinite-dimensional setting. When Y is a Hilbert space, itis true that any compact operator is a limit of finite-rank operators, so that the class of compact operators can bedefined alternatively as the closure in the operator norm of the finite-rank operators. Whether this was true in generalfor Banach spaces (the approximation property) was an unsolved question for many years; in the end Per Enflo gavea counter-example.The origin of the theory of compact operators is in the theory of integral equations, where integral operators supplyconcrete examples of such operators. A typical Fredholm integral equation gives rise to a compact operator K onfunction spaces; the compactness property is shown by equicontinuity. The method of approximation by finite-rankoperators is basic in the numerical solution of such equations. The abstract idea of Fredholm operator is derived fromthis connection.

3.1 Equivalent formulations

A bounded operator T : X → Y is compact if and only if any of the following is true

• Image of the unit ball in X under T is relatively compact in Y.

• Image of any bounded set under T is relatively compact in Y.

• Image of any bounded set under T is totally bounded in Y.

• there exists a neighbourhood of 0, U ⊂ X , and compact set V ⊂ Y such that T (U) ⊂ V .

• For any sequence (xn)n∈N from the unit ball in X, the sequence (Txn)n∈N contains a Cauchy subsequence.

Note that if a linear operator is compact, then it is easy to see that it is bounded, and hence continuous.

3.2 Important properties

In the following, X, Y, Z, W are Banach spaces, B(X, Y) is the space of bounded operators from X to Y with theoperator norm, K(X, Y) is the space of compact operators from X to Y, B(X) = B(X, X), K(X) = K(X, X), idX is theidentity operator on X.

• K(X, Y) is a closed subspace of B(X, Y): Let Tn, n ∈ N, be a sequence of compact operators from one Banachspace to the other, and suppose that Tn converges to T with respect to the operator norm. Then T is alsocompact.

6

3.3. ORIGINS IN INTEGRAL EQUATION THEORY 7

• Conversely, if X, Y are Hilbert spaces, then every compact operator from X to Y is the limit of finite rankoperators. Notably, this is false for general Banach spaces X and Y.

• B(Y,Z) K(X,Y ) B(W,X) ⊆ K(W,Z). In particular, K(X) forms a two-sided operator ideal in B(X).

• idX is compact if and only if X has finite dimension.

• For any T ∈ K(X), idX − T is a Fredholm operator of index 0. In particular, im (idX − T ) is closed. Thisis essential in developing the spectral properties of compact operators. One can notice the similarity betweenthis property and the fact that, if M and N are subspaces of a Banach space where M is closed and N isfinite-dimensional, then M + N is also closed.

• Any compact operator is strictly singular, but not vice versa.[1]

• An operator is compact if and only if its adjoint is compact (Schauder’s theorem).

3.3 Origins in integral equation theory

A crucial property of compact operators is the Fredholm alternative, which asserts that the existence of solution oflinear equations of the form(λK + I)u = f

(where K is a compact operator, f is a given function, and u is the unknown function to be solved for) behaves muchlike as in finite dimensions. The spectral theory of compact operators then follows, and it is due to Frigyes Riesz(1918). It shows that a compact operator K on an infinite-dimensional Banach space has spectrum that is either afinite subset of C which includes 0, or the spectrum is a countably infinite subset of C which has 0 as its only limitpoint. Moreover, in either case the non-zero elements of the spectrum are eigenvalues of K with finite multiplicities(so that K − λI has a finite-dimensional kernel for all complex λ ≠ 0).An important example of a compact operator is compact embedding of Sobolev spaces, which, along with the Gårdinginequality and the Lax–Milgram theorem, can be used to convert an elliptic boundary value problem into a Fredholmintegral equation.[2] Existence of the solution and spectral properties then follow from the theory of compact opera-tors; in particular, an elliptic boundary value problem on a bounded domain has infinitely many isolated eigenvalues.One consequence is that a solid body can vibrate only at isolated frequencies, given by the eigenvalues, and arbitrarilyhigh vibration frequencies always exist.The compact operators from a Banach space to itself form a two-sided ideal in the algebra of all bounded operatorson the space. Indeed, the compact operators on an infinite-dimensional Hilbert space form a maximal ideal, so thequotient algebra, known as the Calkin algebra, is simple.

3.4 Compact operator on Hilbert spaces

Main article: Compact operator on Hilbert space

An equivalent definition of compact operators on a Hilbert space may be given as follows.An operator T on an infinite-dimensional Hilbert spaceH

T : H → H

is said to be compact if it can be written in the form

T =∞∑

n=1

λn⟨fn, ·⟩gn ,

8 CHAPTER 3. COMPACT OPERATOR

where f1, f2, . . . and g1, g2, . . . are (not necessarily complete) orthonormal sets, and λ1, λ2, . . . is a sequence ofpositive numbers with limit zero, called the singular values of the operator. The singular values can accumulate onlyat zero. If the sequence becomes stationary at zero, that is λN+k = 0 for some N ∈ N, and every k = 1, 2, . . . ,then the operator has finite rank, i.e, a finite-dimensional range and can be written as

T =N∑

n=1

λn⟨fn, ·⟩gn .

The bracket ⟨·, ·⟩ is the scalar product on the Hilbert space; the sum on the right hand side converges in the operatornorm.An important subclass of compact operators is the trace-class or nuclear operators.

3.5 Completely continuous operators

Let X and Y be Banach spaces. A bounded linear operator T : X→ Y is called completely continuous if, for everyweakly convergent sequence (xn) from X, the sequence (Txn) is norm-convergent in Y (Conway 1985, §VI.3).Compact operators on a Banach space are always completely continuous. If X is a reflexive Banach space, then everycompletely continuous operator T : X→ Y is compact.

3.6 Examples• Every finite rank operator is compact.

• For ℓp and a sequence (tn) converging to zero, the multiplication operator (Tx)n = tn xn is compact.

• For some fixed g ∈ C([0, 1]; R), define the linear operator T from C([0, 1]; R) to C([0, 1]; R) by

(Tf)(x) =

∫ x

0

f(t)g(t) dt.

That the operator T is indeed compact follows from the Ascoli theorem.

• More generally, if Ω is any domain in Rn and the integral kernel k : Ω × Ω → R is a Hilbert—Schmidt kernel,then the operator T on L2(Ω; R) defined by

(Tf)(x) =

∫Ω

k(x, y)f(y) dy

is a compact operator.

• By Riesz’s lemma, the identity operator is a compact operator if and only if the space is finite-dimensional.

3.7 See also• Spectral theory of compact operators• Fredholm operator• Fredholm integral equations• Fredholm alternative• Compact embedding• Strictly singular operator

3.8. NOTES 9

3.8 Notes[1] N.L. Carothers, A Short Course on Banach Space Theory, (2005) London Mathematical Society Student Texts 64, Cam-

bridge University Press.

[2] William McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000

3.9 References• Conway, John B. (1985). A course in functional analysis. Springer-Verlag. ISBN 3-540-96042-2.

• Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations. Texts inApplied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. 356. ISBN 0-387-00444-0. (Section7.5)

• Kutateladze, S.S. (1996). Fundamentals of Functional Analysis. Texts in Mathematical Sciences 12 (Seconded.). New York: Springer-Verlag. p. 292. ISBN 978-0-7923-3898-7.

Chapter 4

Compact space

“Compactness” redirects here. For the concept in first-order logic, see Compactness theorem.In mathematics, and more specifically 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 fixed distance of each other). Examples include a closed interval, a rectangle, or a finite set ofpoints. This notion is defined for more general topological spaces than Euclidean space in various ways.One such generalization is that a space is sequentially compact if any infinite sequence of points sampled from thespace must frequently (infinitely often) get arbitrarily close to some point of the space. An equivalent definition isthat every sequence of points must have an infinite 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 infinite 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 MauriceFréchet 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

10

4.1. HISTORICAL DEVELOPMENT 11

Arzelà–Ascoli 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, different notions of compactness are notnecessarily equivalent. The most useful notion, which is the standard definition of the unqualified term compactness,is phrased in terms of the existence of finite 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 finite sets. In spaces that are compact inthis sense, it is often possible to patch together information that holds locally—that is, in a neighborhood of eachpoint—into corresponding statements that hold throughout the space, and many theorems are of this character.The term compact set is sometimes a synonym for compact space, but usually refers to a compact subspace of atopological space.

4.1 Historical development

In 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. Bolzano’s 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 infinitely 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 significance of Bolzano’s 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 Bolzano–Weierstrass 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 Arzelà.[2] The culmination oftheir investigations, the Arzelà–Ascoli theorem, was a generalization of the Bolzano–Weierstrass 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 Bolzano’s “limit point”. Towards the beginning of the twentieth century, results similar to that ofArzelà 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 Arzelà–Ascoli theorem held in the sense of mean convergence—or convergence in whatwould later be dubbed a Hilbert space. This ultimately led to the notion of a compact operator as an offshoot of thegeneral notion of a compact space. It was Maurice Fréchet who, in 1906, had distilled the essence of the Bolzano–Weierstrass 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 different 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 defined 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 finite number of these that also covered it. The significance of this lemma wasrecognized by Émile Borel (1895), and it was generalized to arbitrary collections of intervals by Pierre Cousin (1895)and Henri Lebesgue (1904). The Heine–Borel theorem, as the result is now known, is another special propertypossessed by closed and bounded sets of real numbers.This property was significant 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 Heine–Borel 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 Fréchet, now called(relative) sequential compactness, under appropriate conditions followed from the version of compactness that wasformulated in terms of the existence of finite 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.

12 CHAPTER 4. COMPACT SPACE

4.2 Basic examples

An example of a compact space is the (closed) unit interval [0,1] of real numbers. If one chooses an infinite 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 infinite number of points sampled from a disk, some subsetof those points must get arbitrarily close either to a point within the disc, or to a point on the boundary. However, anopen disk is not compact, because a sequence of points can tend to the boundary without getting arbitrarily close toany point in the interior. Likewise, spheres are compact, but a sphere missing a point is not since a sequence of pointscan tend to the missing point, thereby not getting arbitrarily close to any point within the space. Lines and planes arenot compact, since one can take a set of equally-spaced points in any given direction without approaching any point.

4.3 Definitions

Various definitions 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 Bolzano–Weierstrass theorem, that anyinfinite 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 different notions of compactness are not equivalent, and the most usefulnotion of compactness—originally called bicompactness—is defined using covers consisting of open sets (see Opencover definition below). That this form of compactness holds for closed and bounded subsets of Euclidean space isknown as the Heine–Borel theorem. Compactness, when defined in this manner, often allows one to take informationthat is known locally—in a neighbourhood of each point of the space—and to extend it to information that holdsglobally throughout the space. An example of this phenomenon is Dirichlet’s theorem, to which it was originallyapplied by Heine, that a continuous function on a compact interval is uniformly continuous; here, continuity is a localproperty of the function, and uniform continuity the corresponding global property.

4.3.1 Open cover definition

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

Uαα∈A

of open subsets of X such that

X =∪α∈A

Uα,

there is a finite subset J of A such that

X =∪i∈J

Ui.

Some branches of mathematics such as algebraic geometry, typically influenced 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 bothHausdorff and quasi-compact. A compact set is sometimes referred to as a compactum, plural compacta.

4.3. DEFINITIONS 13

4.3.2 Equivalent definitions

Assuming the axiom of choice, the following are equivalent:

1. A topological space X is compact.

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

3. X has a sub-base such that every cover of the space bymembers of the sub-base has a finite subcover (Alexander’ssub-base theorem)

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

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

6. Every filter on X has a convergent refinement.

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

8. Every infinite 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 Heine–Boreltheorem.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 first-countable uniform spaces).

4. (X,d) is limit point compact; that is, every infinite 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 satisfies the following properties:

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

2. (X,d) is second-countable, separable and Lindelöf – these three conditions are equivalent for metric spaces.The converse is not true; e.g., a countable discrete space satisfies 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 finite subcover.It is complete but not totally bounded.

14 CHAPTER 4. COMPACT SPACE

Characterization by continuous functions

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

evp : C(X) → R

given by evp(f)=f(p) is a ring homomorphism. The kernel of evp is a maximal ideal, since the residue field C(X)/kerevp is the field of real numbers, by the first isomorphism theorem. A topological spaceX is pseudocompact if and onlyif every maximal ideal in C(X) has residue field 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 fieldC(X)/m is a (non-archimedean) hyperreal field. 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 infinitely close to a point x0 of X (more precisely, x is contained in the monad of x0).

Hyperreal definition

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

4.3.3 Compactness of subspaces

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

Uαα∈A

of open subsets of X such that

K ⊂∪α∈A

Uα,

there is a finite subset J of A such that

K ⊂∪i∈J

Ui.

4.4 Properties of compact spaces

4.4.1 Functions and compact spaces

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

4.4.2 Compact spaces and set operations

A closed subset of a compact space is compact.,[11] and a finite union of compact sets is compact.

4.5. EXAMPLES 15

The product of any collection of compact spaces is compact. (Tychonoff’s 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 Alexandroff one-point compactification. By the same construction, every locally compact Hausdorff space X isan open dense subspace of a compact Hausdorff space having at most one point more than X.

4.4.3 Ordered compact spaces

A 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 infima).[12]

4.5 Examples• Any finite topological space, including the empty set, is compact. More generally, any space with a finitetopology (only finitely many open sets) is compact; this includes in particular the trivial topology.

• Any space carrying the cofinite topology is compact.

• Any locally compact Hausdorff space can be turned into a compact space by adding a single point to it, bymeans of Alexandroff one-point compactification. The one-point compactification of R is homeomorphic tothe circle S1; the one-point compactification of R2 is homeomorphic to the sphere S2. Using the one-pointcompactification, one can also easily construct compact spaces which are not Hausdorff, by starting with anon-Hausdorff 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, satisfies the property that no uncountable set is compact.

• In the cocountable topology on an uncountable set, no infinite set is compact. Like the previous example, thespace as a whole is not locally compact but is still Lindelöf.

• The closed unit interval [0,1] is compact. This follows from the Heine–Borel 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 finite subcover. Similarly, the set of rational numbers in the closedinterval [0,1] is not compact: the sets of rational numbers in the intervals[0,

1

π− 1

n

]and

[1

π+

1

n, 1

]cover all the rationals in [0, 1] for n = 4, 5, … but this cover does not have a finite 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 finitesubcover. For example, intervals (n−1, n+1) , where n takes all integer values in Z, cover R but there is nofinite subcover.

• For every natural number n, the n-sphere is compact. Again from the Heine–Borel theorem, the closed unitball of any finite-dimensional normed vector space is compact. This is not true for infinite dimensions; in fact,a normed vector space is finite-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.(Alaoglu’s theorem)

16 CHAPTER 4. COMPACT SPACE

• 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 definea topology on K so that a sequence fn in K converges towards f ∈ K if and only if fn(x) 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 Tychonofftheorem.

• 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) = supx∈[0,1]

|f(x)− g(x)|.

Then by Arzelà–Ascoli 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 ℓ2 may have any compact nonemptysubset of C as spectrum.

4.5.1 Algebraic examples

• Compact groups such as an orthogonal group are compact, while groups such as a general linear group are not.

• Since the p-adic integers are homeomorphic to the Cantor set, they form a compact set.

• The spectrum of any commutative ring with the Zariski topology (that is, the set of all prime ideals) is compact,but never Hausdorff (except in trivial cases). In algebraic geometry, such topological spaces are examples ofquasi-compact schemes, “quasi” referring to the non-Hausdorff 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 Hausdorff spaces, form the abstract framework in which these spectraare studied. Such spaces are also useful in the study of profinite groups.

• The structure space of a commutative unital Banach algebra is a compact Hausdorff space.

• The Hilbert cube is compact, again a consequence of Tychonoff’s theorem.

• A profinite group (e.g., Galois group) is compact.

4.6 See also

• Compactly generated space

• Eberlein compactum

• Exhaustion by compact sets

• Lindelöf space

• Metacompact space

• Noetherian space

• Orthocompact space

• Paracompact space

4.7. NOTES 17

4.7 Notes[1] Kline 1972, pp. 952–953; 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] Arkhangel’skii & 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] Arkhangel’skii &Fedorchuk 1990, Theorem 5.2.2; See also Compactness is preserved under a continuousmap at PlanetMath.org.

[10] Arkhangel’skii & Fedorchuk 1990, Corollary 5.2.1

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

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

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

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

• Arkhangel’skii, A.V. (2001), “Compact space”, inHazewinkel, 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 gewähren, 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, Émile (1895), “Sur quelques points de la théorie des fonctions”, Annales Scientifiques de l'École NormaleSupérieure, 3 12: 9–55, JFM 26.0429.03

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

• Arzelà, Cesare (1895), “Sulle funzioni di linee”, Mem. Accad. Sci. Ist. Bologna Cl. Sci. Fis. Mat. 5 (5):55–74.

• Arzelà, Cesare (1882–1883), “Un'osservazione intorno alle serie di funzioni”, Rend. Dell' Accad. R. Delle Sci.Dell'Istituto di Bologna: 142–159.

• Ascoli, G. (1883–1884), “Le curve limiti di una varietà data di curve”, Atti della R. Accad. Dei Lincei Memoriedella Cl. Sci. Fis. Mat. Nat. 18 (3): 521–586.

• Fréchet, Maurice (1906), “Sur quelques points du calcul fonctionnel”, Rendiconti del Circolo Matematico diPalermo 22 (1): 1–72, 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.

18 CHAPTER 4. COMPACT SPACE

• 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), Leçons sur l'intégration 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): 131–147, doi:10.2307/1994440, JSTOR 1994440.

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

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

4.9 External links• Countably compact at PlanetMath.org.

• Sundström, Manya Raman (2010). “A pedagogical history of compactness”. v1. arXiv:1006.4131 [math.HO].

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

Chapter 5

Compactly embedded

In mathematics, the notion of being compactly embedded expresses the idea that one set or space is “well contained”inside another. There are versions of this concept appropriate to general topology and functional analysis.

5.1 Definition (topological spaces)

Let (X, T) be a topological space, and let V and W be subsets of X. We say that V is compactly embedded in W,and write V ⊂⊂W, if

• V ⊆ Cl(V) ⊆ Int(W), where Cl(V) denotes the closure of V, and Int(W) denotes the interior ofW ; and

• Cl(V) is compact.

5.2 Definition (normed spaces)

Let X and Y be two normed vector spaces with norms ||•||X and ||•||Y respectively, and suppose that X ⊆ Y. We saythat X is compactly embedded in Y, and write X ⊂⊂ Y, if

• X is continuously embedded in Y; i.e., there is a constant C such that ||x||Y ≤ C||x||X for all x in X; and

• The embedding of X into Y is a compact operator: any bounded set in X is totally bounded in Y, i.e. everysequence in such a bounded set has a subsequence that is Cauchy in the norm ||•||Y .

If Y is a Banach space, an equivalent definition is that the embedding operator (the identity) i : X → Y is a compactoperator.When applied to functional analysis, this version of compact embedding is usually used with Banach spaces of func-tions. Several of the Sobolev embedding theorems are compact embedding theorems.

5.3 References• Adams, Robert A. (1975). Sobolev Spaces. Boston, MA: Academic Press. ISBN 978-0-12-044150-1..

• Evans, Lawrence C. (1998). Partial differential equations. Providence, RI: American Mathematical Society.ISBN 0-8218-0772-2..

• Renardy, M., & Rogers, R. C. (1992). An Introduction to Partial Differential Equations. Berlin: Springer-Verlag. ISBN 3-540-97952-2..

19

Chapter 6

Cover (topology)

In mathematics, a cover of a set X is a collection of sets whose union contains X as a subset. Formally, if

C = Uα : α ∈ A

is an indexed family of sets Uα , then C is a cover of X if

X ⊆∪α∈A

Uα.

6.1 Cover in topology

Covers are commonly used in the context of topology. If the set X is a topological space, then a cover C of X is acollection of subsets Uα of X whose union is the whole space X. In this case we say that C covers X, or that the setsUα cover X. Also, if Y is a subset of X, then a cover of Y is a collection of subsets of X whose union contains Y, i.e.,C is a cover of Y if

Y ⊆∪α∈A

Uα

Let C be a cover of a topological space X. A subcover of C is a subset of C that still covers X.We say that C is an open cover if each of its members is an open set (i.e. each Uα is contained in T, where T is thetopology on X).A cover of X is said to be locally finite if every point of X has a neighborhood which intersects only finitely many setsin the cover. Formally, C = Uα is locally finite if for any x ∈ X, there exists some neighborhood N(x) of x suchthat the set

α ∈ A : Uα ∩N(x) = ∅

is finite. A cover of X is said to be point finite if every point of X is contained in only finitely many sets in the cover.(locally finite implies point finite)

6.2 Refinement

A refinement of a cover C of a topological space X is a new cover D of X such that every set in D is contained insome set in C. Formally,

20

6.3. COMPACTNESS 21

D = Vβ∈B

is a refinement of

Uα∈A when ∀β ∃α Vβ ⊆ Uα

In other words, there is a refinement map ϕ : B → A satisfying Vβ ⊆ Uϕ(β) for every β ∈ B . This map is used,for instance, in the Čech cohomology of X.[1]

Every subcover is also a refinement, but the opposite is not always true. A subcover is made from the sets that are inthe cover, but omitting some of them; whereas a refinement is made from any sets that are subsets of the sets in thecover.The refinement relation is a preorder on the set of covers of X.Generally speaking, a refinement of a given structure is another that in some sense contains it. Examples are to befound when partitioning an interval (one refinement of a0 < a1 < ... < an being a0 < b0 < a1 < a2 < ... < an <b1 ), considering topologies (the standard topology in euclidean space being a refinement of the trivial topology).When subdividing simplicial complexes (the first barycentric subdivision of a simplicial complex is a refinement), thesituation is slightly different: every simplex in the finer complex is a face of some simplex in the coarser one, andboth have equal underlying polyhedra.Yet another notion of refinement is that of star refinement.

6.3 Compactness

The language of covers is often used to define several topological properties related to compactness. A topologicalspace X is said to be

• Compact, if every open cover has a finite subcover, (or equivalently that every open cover has a finite refinement);

• Lindelöf, if every open cover has a countable subcover, (or equivalently that every open cover has a countablerefinement);

• Metacompact, if every open cover has a point finite open refinement;

• Paracompact, if every open cover admits a locally finite open refinement.

For some more variations see the above articles.

6.4 Covering dimension

A topological space X is said to be of covering dimension n if every open cover of X has a point finite open refinementsuch that no point of X is included in more than n+1 sets in the refinement and if n is the minimum value for whichthis is true.[2] If no such minimal n exists, the space is said to be of infinite covering dimension.

6.5 See also

• Covering space

• Atlas (topology)

• Set cover problem

22 CHAPTER 6. COVER (TOPOLOGY)

6.6 Notes[1] Bott, Tu (1982). Differential Forms in Algebraic Topology. p. 111.

[2] Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

6.7 References1. Introduction to Topology, Second Edition, Theodore W. Gamelin & Robert Everist Greene. Dover Publications

1999. ISBN 0-486-40680-6

2. General Topology, John L. Kelley. D. Van Nostrand Company, Inc. Princeton, NJ. 1955.

6.8 External links• Hazewinkel, Michiel, ed. (2001), “Covering (of a set)", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 7

Exhaustion by compact sets

In mathematics, especially analysis, exhaustion by compact sets of an open set E in the Euclidean space Rn (or amanifold with countable base) is an increasing sequence of compact sets Kj , where by increasing we meanKj is asubset ofKj+1 , with the limit (union) of the sequence being E.Sometimes one requires the sequence of compact sets to satisfy one more property— that Kj is contained in theinterior ofKj+1 for each j . This, however, is dispensed in Rn or a manifold with countable base.For example, consider a unit open disk and the concentric closed disk of each radius inside. That is letE = z; |z| <1 andKj = z; |z| ≤ (1− 1/j) . Then taking the limit (union) of the sequenceKj gives E. The example can beeasily generalized in other dimensions.

7.1 See also• σ-compact space

7.2 References• Leon Ehrenpreis, Theory of Distributions for Locally Compact Spaces, American Mathematical Society, 1982.ISBN 0-8218-1221-1.

7.3 External links• Exhaustion by compact sets at PlanetMath.org.

23

Chapter 8

Feebly compact space

In mathematics, a topological space is feebly compact if every locally finite cover by nonempty open sets is finite.Some facts:

• Every compact space is feebly compact.

• Every feebly compact paracompact space is compact.

• Every feebly compact space is pseudocompact but the converse is not necessarily true.

• For a completely regular Hausdorff space the properties of being feebly compact and pseudocompact are equiv-alent.

• Any maximal feebly compact space is submaximal.

24

Chapter 9

Functional analysis

For the assessment and treatment of human behavior, see Functional analysis (psychology).Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces

One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator ona function space, a common construction in functional analysis.

endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operatorsacting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysislie in the study of spaces of functions and the formulation of properties of transformations of functions such as theFourier transform as transformations defining continuous, unitary etc. operators between function spaces. This pointof view turned out to be particularly useful for the study of differential and integral equations.The usage of the word functional goes back to the calculus of variations, implying a function whose argument is afunction and the name was first used in Hadamard's 1910 book on that subject. However, the general concept ofa functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra. Thetheory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. Hadamardalso founded the modern school of linear functional analysis further developed by Riesz and the group of Polishmathematicians around Stefan Banach.In modern introductory texts to functional analysis, the subject is seen as the study of vector spaces endowed witha topology, in particular infinite dimensional spaces. In contrast, linear algebra deals mostly with finite dimensionalspaces, and does not use topology. An important part of functional analysis is the extension of the theory of measure,integration, and probability to infinite dimensional spaces, also known as infinite dimensional analysis.

25

26 CHAPTER 9. FUNCTIONAL ANALYSIS

9.1 Normed vector spaces

The basic and historically first class of spaces studied in functional analysis are complete normed vector spaces overthe real or complex numbers. Such spaces are called Banach spaces. An important example is a Hilbert space, wherethe norm arises from an inner product. These spaces are of fundamental importance in many areas, including themathematical formulation of quantum mechanics.More generally, functional analysis includes the study of Fréchet spaces and other topological vector spaces notendowed with a norm.An important object of study in functional analysis are the continuous linear operators defined on Banach and Hilbertspaces. These lead naturally to the definition of C*-algebras and other operator algebras.

9.1.1 Hilbert spaces

Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinalityof the orthonormal basis. Finite-dimensional Hilbert spaces are fully understood in linear algebra, and infinite-dimensional separable Hilbert spaces are isomorphic to ℓ 2(ℵ0) . Separability being important for applications,functional analysis of Hilbert spaces consequently mostly deals with this space. One of the open problems in func-tional analysis is to prove that every bounded linear operator on a Hilbert space has a proper invariant subspace. Manyspecial cases of this invariant subspace problem have already been proven.

9.1.2 Banach spaces

General Banach spaces are more complicated than Hilbert spaces, and cannot be classified in such a simple manneras those. In particular, many Banach spaces lack a notion analogous to an orthonormal basis.Examples of Banach spaces are L p -spaces for any real number p ≥ 1 . Given also a measure µ on set X , thenL p(X) , sometimes also denoted L p(X,µ) or L p(µ) , has as its vectors equivalence classes [ f ] of measurablefunctions whose absolute value's p -th power has finite integral, that is, functions f for which one has

∫X

|f(x)|p dµ(x) < +∞

If µ is the counting measure, then the integral may be replaced by a sum. That is, we require

∑x∈X

|f(x)|p < +∞

Then it is not necessary to deal with equivalence classes, and the space is denoted ℓ p(X) , written more simply ℓ pin the case when X is the set of non-negative integers.In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear maps from thespace into its underlying field, so-called functionals. A Banach space can be canonically identified with a subspace ofits bidual, which is the dual of its dual space. The corresponding map is an isometry but in general not onto. A generalBanach space and its bidual need not even be isometrically isomorphic in any way, contrary to the finite-dimensionalsituation. This is explained in the dual space article.Also, the notion of derivative can be extended to arbitrary functions between Banach spaces. See, for instance, theFréchet derivative article.

9.2 Major and foundational results

Important results of functional analysis include:

9.2. MAJOR AND FOUNDATIONAL RESULTS 27

9.2.1 Uniform boundedness principle

Main article: Banach-Steinhaus theorem

The uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functionalanalysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cor-nerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus boundedoperators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operatornorm.The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus but it was also proven independentlyby Hans Hahn.

Theorem (Uniform Boundedness Principle). Let X be a Banach space and Y be a normed vectorspace. Suppose that F is a collection of continuous linear operators from X to Y. If for all x in X one has

supT∈F ∥T (x)∥Y <∞,

then

supT∈F ∥T∥B(X,Y ) <∞.

9.2.2 Spectral theorem

Main article: Spectral theorem

There are many theorems known as the spectral theorem, but one in particular has many applications in functionalanalysis. Let A be the operator of multiplication by t on L2[0, 1], that is

[Aφ](t) = tφ(t).

Theorem:[1] Let A be a bounded self-adjoint operator on a Hilbert space H. Then there is a measure space (X, Σ, μ)and a real-valued essentially bounded measurable function f on X and a unitary operator U:H → L2μ(X) such that

U∗TU = A

where T is the multiplication operator:

[Tφ](x) = f(x)φ(x).

and ∥T∥ = ∥f∥∞This is the beginning of the vast research area of functional analysis called operator theory; see also the spectralmeasure.There is also an analogous spectral theorem for bounded normal operators on Hilbert spaces. The only difference inthe conclusion is that now f may be complex-valued.

9.2.3 Hahn-Banach theorem

Main article: Hahn-Banach theorem

TheHahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionalsdefined on a subspace of some vector space to the whole space, and it also shows that there are “enough” continuouslinear functionals defined on every normed vector space to make the study of the dual space “interesting”.

28 CHAPTER 9. FUNCTIONAL ANALYSIS

Hahn–Banach Theorem:[2] If p : V → R is a sublinear function, and φ : U → R is a linear functional on a linearsubspace U ⊆ V which is dominated by p on U, i.e.

φ(x) ≤ p(x) ∀x ∈ U

then there exists a linear extension ψ : V → R of φ to the whole space V, i.e., there exists a linear functional ψ suchthat

ψ(x) = φ(x) ∀x ∈ U,

ψ(x) ≤ p(x) ∀x ∈ V.

9.2.4 Open mapping theorem

Main article: Open mapping theorem (functional analysis)

The open mapping theorem, also known as the Banach–Schauder theorem (named after Stefan Banach and JuliuszSchauder), is a fundamental result which states that if a continuous linear operator between Banach spaces is surjectivethen it is an open map. More precisely,:[2]

Open Mapping Theorem. If X and Y are Banach spaces and A : X → Y is a surjective continuouslinear operator, then A is an open map (i.e. if U is an open set in X, then A(U) is open in Y).

The proof uses the Baire category theorem, and completeness of both X and Y is essential to the theorem. Thestatement of the theorem is no longer true if either space is just assumed to be a normed space, but is true if X andY are taken to be Fréchet spaces.

9.2.5 Closed graph theorem

Main article: Closed graph theorem

The closed graph theorem states the following: If X is a topological space and Y is a compact Hausdorff space, thenthe graph of T is closed if and only if T is continuous.[3]

9.2.6 Other topics

List of functional analysis topics.

9.3 Foundations of mathematics considerations

Most spaces considered in functional analysis have infinite dimension. To show the existence of a vector space basisfor such spaces may require Zorn’s lemma. However, a somewhat different concept, Schauder basis, is usually morerelevant in functional analysis. Many very important theorems require the Hahn–Banach theorem, usually provedusing axiom of choice, although the strictly weaker Boolean prime ideal theorem suffices. The Baire category theorem,needed to prove many important theorems, also requires a form of axiom of choice.

9.4 Points of view

Functional analysis in its present form includes the following tendencies:

9.5. SEE ALSO 29

• Abstract analysis. An approach to analysis based on topological groups, topological rings, and topologicalvector spaces.

• Geometry of Banach spaces contains many topics. One is combinatorial approach connected with Jean Bour-gain; another is a characterization of Banach spaces in which various forms of the law of large numbers hold.

• Noncommutative geometry. Developed by Alain Connes, partly building on earlier notions, such as GeorgeMackey's approach to ergodic theory.

• Connection with quantummechanics. Either narrowly defined as inmathematical physics, or broadly interpretedby, e.g. Israel Gelfand, to include most types of representation theory.

9.5 See also

• List of functional analysis topics

• Spectral theory

9.6 References[1] Hall, B.C. (2013), Quantum Theory for Mathematicians, Springer, p. 147

[2] Rudin, Walter (1991). Functional analysis. McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5.

[3] Munkres, James (2000), Topology (2nd ed.), Upper Saddle River: Prentice Hall, pp. 163–172, ISBN 0-13-181629-2, p.171

9.7 Further reading

• Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd ed., Springer 2007,ISBN 978-3-540-32696-0. Online doi:10.1007/3-540-29587-9 (by subscription)

• Bachman, G., Narici, L.: Functional analysis, Academic Press, 1966. (reprint Dover Publications)

• Banach S. Theory of Linear Operations. Volume 38, North-Holland Mathematical Library, 1987, ISBN 0-444-70184-2

• Brezis, H.: Analyse Fonctionnelle, Dunod ISBN 978-2-10-004314-9 or ISBN 978-2-10-049336-4

• Conway, J. B.: A Course in Functional Analysis, 2nd edition, Springer-Verlag, 1994, ISBN 0-387-97245-5

• Dunford, N. and Schwartz, J.T.: Linear Operators, General Theory, John Wiley & Sons, and other 3 volumes,includes visualization charts

• Edwards, R. E.: Functional Analysis, Theory and Applications, Hold, Rinehart and Winston, 1965.

• Eidelman, Yuli, Vitali Milman, and Antonis Tsolomitis: Functional Analysis: An Introduction, AmericanMathematical Society, 2004.

• Freidman, A.: Foundations of Modern Analysis, Dover Publications, Paperback Edition, July 21, 2010

• Giles,J.R.: Introduction to the Analysis of Normed Linear Spaces,Cambridge University Press,2000

• Hirsch F., Lacombe G. - “Elements of Functional Analysis”, Springer 1999.

• Hutson, V., Pym, J.S., Cloud M.J.: Applications of Functional Analysis and Operator Theory, 2nd edition,Elsevier Science, 2005, ISBN 0-444-51790-1

• Kantorovitz, S.,Introduction to Modern Analysis, Oxford University Press,2003,2nd ed.2006.

30 CHAPTER 9. FUNCTIONAL ANALYSIS

• Kolmogorov, A.N and Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis, DoverPublications, 1999

• Kreyszig, E.: Introductory Functional Analysis with Applications, Wiley, 1989.

• Lax, P.: Functional Analysis, Wiley-Interscience, 2002, ISBN 0-471-55604-1

• Lebedev, L.P. and Vorovich, I.I.: Functional Analysis in Mechanics, Springer-Verlag, 2002

• Michel, Anthony N. and Charles J. Herget: Applied Algebra and Functional Analysis, Dover, 1993.

• Pietsch, Albrecht: History of Banach spaces and linear operators, Birkhauser Boston Inc., 2007, ISBN 978-0-8176-4367-6

• Reed, M., Simon, B.: “Functional Analysis”, Academic Press 1980.

• Riesz, F. and Sz.-Nagy, B.: Functional Analysis, Dover Publications, 1990

• Rudin, W.: Functional Analysis, McGraw-Hill Science, 1991

• Schechter, M.: Principles of Functional Analysis, AMS, 2nd edition, 2001

• Shilov, Georgi E.: Elementary Functional Analysis, Dover, 1996.

• Sobolev, S.L.: Applications of Functional Analysis in Mathematical Physics, AMS, 1963

• Yosida, K.: Functional Analysis, Springer-Verlag, 6th edition, 1980

• Vogt, D., Meise, R.: Introduction to Functional Analysis, Oxford University Press, 1997.

9.8 External links• Hazewinkel, Michiel, ed. (2001), “Functional analysis”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Topics in Real and Functional Analysis by Gerald Teschl, University of Vienna.

• Lecture Notes on Functional Analysis by Yevgeny Vilensky, New York University.

• Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis by John Aldrich Universityof Southampton.

• Lecture videos on functional analysis by Greg Morrow from University of Colorado Colorado Springs

• An Introduction to Functional Analysis on Coursera by John Cagnol from Ecole Centrale Paris

Chapter 10

H-closed space

In mathematics, a topological space X is said to be H-closed, or Hausdorff closed, or absolutely closed if it isclosed in every Hausdorff space containing it as a subspace. This property is a generalization of compactness, sincea compact subset of a Hausdorff space is closed. Thus, every compact Hausdorff space is H-closed. The notion ofan H-closed space has been introduced in 1924 by P. Alexandroff and P. Urysohn.

10.1 Examples and equivalent formulations• The unit interval [0, 1] , endowed with the smallest topology which refines the euclidean topology, and containsQ ∩ [0, 1] as an open set is H-closed but not compact.

• Every regular Hausdorff H-closed space is compact.

• A Hausdorff space is H-closed if and only if every open cover has a finite subfamily with dense union.

10.2 See also• Compact space

10.3 References• K.P. Hart, Jun-iti Nagata, J.E. Vaughan (editors), Encyclopedia of General Topology, Chapter d20 (by JackPorter and Johannes Vermeer)

31

Chapter 11

Hemicompact space

In mathematics, in the field of topology, a topological space is said to be hemicompact if it has a sequence of compactsubsets such that every compact subset of the space lies inside some compact set in the sequence. Clearly, this forcesthe union of the sequence to be the whole space, because every point is compact and hence must lie in one of thecompact sets.

11.1 Examples• Every compact space is hemicompact.

• The real line is hemicompact.

• Every locally compact Lindelöf space is hemicompact.

11.2 Properties

Every hemicompact space is σ-compact and if in addition it is first countable then it is locally compact.If X is a hemicompact space, then the space C(X,M) of all continuous functions f : X → M to a metric space(M, δ) with the compact-open topology is metrizable. To see this, take a sequence K1,K2, . . . of compact subsetsof X such that every compact subset of X lies inside some compact set in this sequence (the existence of such asequence follows from the hemicompactness ofX ). Denote

dn(f, g) = supx∈Kn

δ(f(x), g(x))

for f, g ∈ C(X,M) and n ∈ N . Then

d(f, g) =∞∑

n=1

1

2n· dn(f, g)

1 + dn(f, g)

defines a metric on C(X,M) which induces the compact-open topology.

11.3 See also• Compact space

• Locally compact space

• Lindelöf space

32

11.4. REFERENCES 33

11.4 References• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

Chapter 12

Interior (topology)

S

x

y

The point x is an interior point of S. The point y is on the boundary of S.

In mathematics, specifically in topology, the interior of a subset S of points of a topological space X consists of allpoints of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S.The interior of S is the complement of the closure of the complement of S. In this sense interior and closure are dual

34

12.1. DEFINITIONS 35

notions.The exterior of a set is the interior of its complement, equivalently the complement of its closure; it consists of thepoints that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partitionthe whole space into three blocks (or fewer when one or more of these is empty). The interior and exterior are alwaysopen while the boundary is always closed. Sets with empty interior have been called boundary sets.[1]

12.1 Definitions

12.1.1 Interior point

If S is a subset of a Euclidean space, then x is an interior point of S if there exists an open ball centered at x which iscompletely contained in S. (This is illustrated in the introductory section to this article.)This definition generalizes to any subset S of a metric space X with metric d: x is an interior point of S if there existsr > 0, such that y is in S whenever the distance d(x, y) < r.This definition generalises to topological spaces by replacing “open ball” with "open set". Let S be a subset of atopological space X. Then x is an interior point of S if x is contained in an open subset of S. (Equivalently, x is aninterior point of S if there exists a neighbourhood of x which is contained in S.)

12.1.2 Interior of a set

The interior of a set S is the set of all interior points of S. The interior of S is denoted int(S), Int(S), or So. Theinterior of a set has the following properties.

• int(S) is an open subset of S.

• int(S) is the union of all open sets contained in S.

• int(S) is the largest open set contained in S.

• A set S is open if and only if S = int(S).

• int(int(S)) = int(S) (idempotence).

• If S is a subset of T, then int(S) is a subset of int(T).

• If A is an open set, then A is a subset of S if and only if A is a subset of int(S).

Sometimes the second or third property above is taken as the definition of the topological interior.Note that these properties are also satisfied if “interior”, “subset”, “union”, “contained in”, “largest” and “open” arereplaced by “closure”, “superset”, “intersection”, “which contains”, “smallest”, and “closed”, respectively. For moreon this matter, see interior operator below.

12.2 Examples

M

( )a-ε a+εa

( [ )) [ ]

a is an interior point of M, because there is an ε-neighbourhood of a which is a subset of M.

• In any space, the interior of the empty set is the empty set.

36 CHAPTER 12. INTERIOR (TOPOLOGY)

• In any space X, if A ⊂ X , int(A) is contained in A.• If X is the Euclidean space R of real numbers, then int([0, 1]) = (0, 1).• If X is the Euclidean space R , then the interior of the set Q of rational numbers is empty.• If X is the complex plane C = R2 , then int (z ∈ C : |z| ≤ 1) = z ∈ C : |z| < 1.

• In any Euclidean space, the interior of any finite set is the empty set.

On the set of real numbers one can put other topologies rather than the standard one.

• If X = R , where R has the lower limit topology, then int([0, 1]) = [0, 1).• If one considers on R the topology in which every set is open, then int([0, 1]) = [0, 1].• If one considers on R the topology in which the only open sets are the empty set and R itself, then int([0, 1])is the empty set.

These examples show that the interior of a set depends upon the topology of the underlying space. The last twoexamples are special cases of the following.

• In any discrete space, since every set is open, every set is equal to its interior.• In any indiscrete space X, since the only open sets are the empty set and X itself, we have int(X) = X and forevery proper subset A of X, int(A) is the empty set.

12.3 Interior operator

The interior operator o is dual to the closure operator —, in the sense that

So = X \ (X \ S)—,

and also

S— = X \ (X \ S)o

where X is the topological space containing S, and the backslash refers to the set-theoretic difference.Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be easily translated intothe language of interior operators, by replacing sets with their complements.

12.4 Exterior of a set

Main article: Exterior (topology)

The exterior of a subset S of a topological space X, denoted ext(S) or Ext(S), is the interior int(X \ S) of its relativecomplement. Alternatively, it can be defined as X \ S—, the complement of the closure of S. Many properties followin a straightforward way from those of the interior operator, such as the following.

• ext(S) is an open set that is disjoint with S.• ext(S) is the union of all open sets that are disjoint with S.• ext(S) is the largest open set that is disjoint with S.• If S is a subset of T, then ext(S) is a superset of ext(T).

Unlike the interior operator, ext is not idempotent, but the following holds:

• ext(ext(S)) is a superset of int(S).

12.5. INTERIOR-DISJOINT SHAPES 37

12.5 Interior-disjoint shapes

The red shapes are not interior-disjoint the blue Triangle. The green and the yellow shapes are interior-disjoint with the blue Triangle,but only the yellow shape is entirely disjoint from the blue Triangle.

Two shapes a and b are called interior-disjoint if the intersection of their interiors is empty. Interior-disjoint shapesmay or may not intersect in their boundary.

12.6 See also

• Algebraic interior

• Interior algebra

• Jordan curve theorem

• Quasi-relative interior

• Relative interior

12.7 References[1] Kuratowski, Kazimierz (1922). “Sur l'Operation Ā de l'Analysis Situs” (PDF). FundamentaMathematicae (Warsaw: Polish

Academy of Sciences) 3: 182–199. ISSN 0016-2736.

38 CHAPTER 12. INTERIOR (TOPOLOGY)

12.8 External links• Interior at PlanetMath.org.

Chapter 13

k-cell (mathematics)

A k-cell is a higher-dimensional version of a rectangle or rectangular solid. It is the Cartesian product of k closedintervals on the real line.[1] This essentially means that is a k-dimensional rectangular solid, with each of its edgesbeing equal to one of the closed intervals used in the definition. The k intervals need not be identical. For example,a 2-cell is a rectangle in R2 such that the sides of the rectangles are parallel to the coordinate axes.

13.1 Formal definition

Let a ∈ R and b ∈ R. If ai < bi for all i = 1,...,k, the set of all points x = (x1,...,xk) in Rk whose coordinates satisfy theinequalities aᵢ ≤ x ≤ bi is a k-cell.[2] Every k-cell is compact.[3]

13.2 Intuition

A k-cell of dimension k ≤ 3 is especially simple. For example, a 1-cell is simply the interval [a,b] with a < b. A 2-cellis the rectangle formed by the Cartesian product of two closed intervals, and a 3-cell is a rectangular solid.Note that the sides and edges of a k-cell need not be equal in (Euclidean) length; although the unit cube (which hasboundaries of equal Euclidean length) is a 3-cell, the set of all 3-cells with equal-length edges is a strict subset of theset of all 3-cells.

13.3 References[1] Foran, James (1991-01-07). Fundamentals of Real Analysis. CRC Press. pp. 24–. ISBN 9780824784539. Retrieved 23

May 2014.

[2] Rudin, W: Principles of Mathematical Analysis, page 31. McGraw-Hill, 1976.

[3] Rudin, W: Principles of Mathematical Analysis, page 39. McGraw-Hill, 1976.

39

40 CHAPTER 13. K-CELL (MATHEMATICS)

Projections of K-cells onto the plane (from k=1 to 6. Only the edges of the higher-dimensional cells are shown.

Chapter 14

Lebesgue covering dimension

In mathematics, theLebesgue covering dimension or topological dimension of a topological space is one of severaldifferent ways of defining the dimension of the space in a topologically invariant way.

14.1 Definition

The first formal definition of covering dimension was given by Eduard Čech, based on an earlier result of HenriLebesgue.[1]

A modern definition is as follows. An open cover of a topological space X is a family of open sets whose union is X.The ply of a cover is the smallest number n (if it exists) such that each point of the space belongs to at most n setsin the cover. A refinement of a cover C is another cover, each of whose sets is a subset of a set in C; its ply may besmaller than, or possibly larger than, the ply of C. The covering dimension of a topological space X is defined to bethe minimum value of n, such that every finite open cover C of X has a refinement with ply at most n + 1. If no suchminimal n exists, the space is said to be of infinite covering dimension.As a special case, a topological space is zero-dimensional with respect to the covering dimension if every open coverof the space has a refinement consisting of disjoint open sets so that any point in the space is contained in exactly oneopen set of this refinement.

14.2 Examples

Any given open cover of the unit circle will have a refinement consisting of a collection of open arcs. The circle hasdimension one, by this definition, because any such cover can be further refined to the stage where a given point xof the circle is contained in at most two open arcs. That is, whatever collection of arcs we begin with, some can bediscarded or shrunk, such that the remainder still covers the circle but with simple overlaps.Similarly, any open cover of the unit disk in the two-dimensional plane can be refined so that any point of the diskis contained in no more than three open sets, while two are in general not sufficient. The covering dimension of thedisk is thus two.More generally, the n-dimensional Euclidean space En has covering dimension n.A non-technical illustration of these examples below.

14.3 Properties

• Homeomorphic spaces have the same covering dimension. That is, the covering dimension is a topologicalinvariant.

• The Lebesgue covering dimension coincides with the affine dimension of a finite simplicial complex; this is theLebesgue covering theorem.

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42 CHAPTER 14. LEBESGUE COVERING DIMENSION

• The covering dimension of a normal space is less than or equal to the large inductive dimension.

• Covering dimension of a normal space X is ≤ n if and only if for any closed subset A of X, if f : A→ Sn iscontinuous, then there is an extension of f to g : X → Sn . Here, Sn is the n dimensional sphere.

• (Ostrand’s theorem on colored dimension.) A normal space X satisfies the inequality 0 ≤ dimX ≤ n if andonly if for every locally finite open cover U = Uαα∈A of the space X there exists an open cover V of thespace X which can be represented as the union of n+ 1 families V1,V2, . . . ,Vn+1 , where Vi = Vi,αα∈A, such that each Vi contains disjoint sets and Vi,α ⊂ Uα for each i and α .

• The covering dimension of a paracompact Hausdorff spaceX is greater or equal to its cohomological dimension(in the sense of sheaves),[2] that is one hasHi(X,A) = 0 for every sheaf A of abelian groups onX and everyi larger than the covering dimension of X .

14.4 See also• Dimension theory

• Metacompact space

• Point-finite collection

14.5 Further reading

14.5.1 Historical

• Karl Menger, General Spaces and Cartesian Spaces, (1926) Communications to the Amsterdam Academy ofSciences. English translation reprinted in Classics on Fractals, Gerald A.Edgar, editor, Addison-Wesley (1993)ISBN 0-201-58701-7

• Karl Menger, Dimensionstheorie, (1928) B.G Teubner Publishers, Leipzig.

• A. R. Pears, Dimension Theory of General Spaces, (1975) Cambridge University Press. ISBN 0-521-20515-8

14.5.2 Modern

• V.V. Fedorchuk, The Fundamentals of Dimension Theory, appearing in Encyclopaedia of Mathematical Sci-ences, Volume 17, General Topology I, (1993) A. V. Arkhangel’skii and L. S. Pontryagin (Eds.), Springer-Verlag, Berlin ISBN 3-540-18178-4.

14.6 References[1] Kuperberg, Krystyna, ed. (1995), Collected Works of Witold Hurewicz, American Mathematical Society, Collected works

series 4, American Mathematical Society, p. xxiii, footnote 3, ISBN 9780821800119, Lebesgue’s discovery led later tothe introduction by E. Čech of the covering dimension.

[2] Godement 1973, II.5.12, p. 236

• Godement, Roger (1973), Topologie algébrique et théorie des faisceaux, Paris: Hermann, MR 0345092

14.7 External links• Hazewinkel, Michiel, ed. (2001), “Lebesgue dimension”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 15

Limit point compact

In mathematics, a topological space X is said to be limit point compact[1] or weakly countably compact if everyinfinite subset of X has a limit point in X. This property generalizes a property of compact spaces. In a metric space,limit point compactness, compactness, and sequential compactness are all equivalent. For general topological spaces,however, these three notions of compactness are not equivalent.

15.1 Properties and Examples• Limit point compactness is equivalent to countable compactness ifX is a T1-space and is equivalent to compactnessif X is a metric space.

• An example of a space X that is not weakly countably compact is any countable (or larger) set with the discretetopology. A more interesting example is the countable complement topology.

• Even though a continuous function from a compact space X, to an ordered set Y in the order topology, mustbe bounded, the same thing does not hold if X is limit point compact. An example is given by the spaceX ×Z(where X = 1, 2 carries the indiscrete topology and Z is the set of all integers carrying the discrete topology)and the function f = πZ given by projection onto the second coordinate. Clearly, ƒ is continuous and X × Zis limit point compact (in fact, every nonempty subset ofX ×Z has a limit point) but ƒ is not bounded, and infact f(X × Z) = Z is not even limit point compact.

• Every countably compact space (and hence every compact space) is weakly countably compact, but the converseis not true.

• For metrizable spaces, compactness, limit point compactness, and sequential compactness are all equivalent.

• The set of all real numbers, R, is not limit point compact; the integers are an infinite set but do not have a limitpoint in R.

• If (X, T) and (X, T*) are topological spaces with T* finer than T and (X, T*) is limit point compact, then so is(X, T).

• A finite space is vacuously limit point compact.

15.2 See also• Compact space

• Sequential compactness

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44 CHAPTER 15. LIMIT POINT COMPACT

• Metric space

• Bolzano-Weierstrass theorem

• Countably compact space

15.3 Notes[1] The terminology “limit point compact” appears in a topology textbook by James Munkres, and is apparently due to him.

According to him, some call the property "Fréchet compactness”, while others call it the "Bolzano-Weierstrass property".Munkres, p. 178–179.

15.4 References• James Munkres (1999). Topology (2nd edition ed.). Prentice Hall. ISBN 0-13-181629-2.

• This article incorporates material from Weakly countably compact on PlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

Chapter 16

Lindelöf space

In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. TheLindelöf property is a weakening of the more commonly used notion of compactness, which requires the existenceof a finite subcover.A strongly Lindelöf space is a topological space such that every open subspace is Lindelöf. Such spaces are alsoknown as hereditarily Lindelöf spaces, because all subspaces of such a space are Lindelöf.Lindelöf spaces are named for the Finnish mathematician Ernst Leonard Lindelöf.

16.1 Properties of Lindelöf spaces

In general, no implications hold (in either direction) between the Lindelöf property and other compactness properties,such as paracompactness. But by the Morita theorem, every regular Lindelöf space is paracompact.Any second-countable space is a Lindelöf space, but not conversely. However, the matter is simpler for metric spaces.A metric space is Lindelöf if and only if it is separable, and if and only if it is second-countable.An open subspace of a Lindelöf space is not necessarily Lindelöf. However, a closed subspace must be Lindelöf.Lindelöf is preserved by continuous maps. However, it is not necessarily preserved by products, not even by finiteproducts.A Lindelöf space is compact if and only if it is countably compact.Any σ-compact space is Lindelöf.

16.2 Properties of strongly Lindelöf spaces• Any second-countable space is a strongly Lindelöf space

• Any Suslin space is strongly Lindelöf.

• Strongly Lindelöf spaces are closed under taking countable unions, subspaces, and continuous images.

• Every Radon measure on a strongly Lindelöf space is moderated.

16.3 Product of Lindelöf spaces

The product of Lindelöf spaces is not necessarily Lindelöf. The usual example of this is the Sorgenfrey plane S ,which is the product of the real line R under the half-open interval topology with itself. Open sets in the Sorgenfreyplane are unions of half-open rectangles that include the south and west edges and omit the north and east edges,including the northwest, northeast, and southeast corners. The antidiagonal of S is the set of points (x, y) such thatx+ y = 0 .

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46 CHAPTER 16. LINDELÖF SPACE

Consider the open covering of S which consists of:

1. The set of all rectangles (−∞, x)× (−∞, y) , where (x, y) is on the antidiagonal.

2. The set of all rectangles [x,+∞)× [y,+∞) , where (x, y) is on the antidiagonal.

The thing to notice here is that each point on the antidiagonal is contained in exactly one set of the covering, so allthese sets are needed.Another way to see that S is not Lindelöf is to note that the antidiagonal defines a closed and uncountable discretesubspace of S . This subspace is not Lindelöf, and so the whole space cannot be Lindelöf as well (as closed subspacesof Lindelöf spaces are also Lindelöf).The product of a Lindelöf space and a compact space is Lindelöf.

16.4 Generalisation

The following definition generalises the definitions of compact and Lindelöf: a topological space is κ -compact (or κ-Lindelöf), where κ is any cardinal, if every open cover has a subcover of cardinality strictly less than κ . Compact isthen ℵ0 -compact and Lindelöf is then ℵ1 -compact.The Lindelöf degree, or Lindelöf number l(X) , is the smallest cardinal κ such that every open cover of the spaceXhas a subcover of size at most κ . In this notation, X is Lindelöf if l(X) = ℵ0 . The Lindelöf number as definedabove does not distinguish between compact spaces and Lindelöf non compact spaces. Some authors gave the nameLindelöf number to a different notion: the smallest cardinal κ such that every open cover of the spaceX has a subcoverof size strictly less than κ .[1] In this latter (and less used) sense the Lindelöf number is the smallest cardinal κ suchthat a topological space X is κ -compact. This notion is sometimes also called the compactness degree of the spaceX .[2]

16.5 See also• Axioms of countability

• Lindelöf’s lemma

16.6 Notes[1] Mary Ellen Rudin, Lectures on set theoretic topology, Conference Board of the Mathematical Sciences, American Math-

ematical Society, 1975, p. 4, retrievable on Google Books

[2] Hušek,Miroslav (1969), “The class of k-compact spaces is simple”,Mathematische Zeitschrift 110: 123–126, doi:10.1007/BF01124977,MR 0244947.

16.7 References• Michael Gemignani, Elementary Topology (ISBN 0-486-66522-4) (see especially section 7.2)

• 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.

• I. Juhász (1980). Cardinal functions in topology - ten years later. Math. Centre Tracts, Amsterdam. ISBN90-6196-196-3.

• Munkres, James. Topology, 2nd ed.

• http://arxiv.org/abs/1301.5340 Generalized Lob’s Theorem.Strong Reflection Principles and Large CardinalAxioms.Consistency Results in Topology

Chapter 17

Locally compact space

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking,each small portion of the space looks like a small portion of a compact space.

17.1 Formal definition

Let X be a topological space. Most commonly X is called locally compact, if every point of X has a compactneighbourhood.There are other common definitions: They are all equivalent if X is a Hausdorff space (or preregular). But theyare not equivalent in general:

1. every point of X has a compact neighbourhood.2. every point of X has a closed compact neighbourhood.2′. every point of X has a relatively compact neighbourhood.2″. every point of X has a local base of relatively compact neighbourhoods.3. every point of X has a local base of compact neighbourhoods.3′. for every point x of X, every neighbourhood of x contains a compact neighbourhood of x.4. X is Hausdorff and satisfies any (all) of the previous conditions.

Logical relations among the conditions:

• Conditions (2), (2′), (2″) are equivalent.

• Conditions (3), (3′) are equivalent.

• Neither of conditions (2), (3) implies the other.

• Each condition implies (1).

• Compactness implies conditions (1) and (2), but not (3).

Condition (1) is probably the most commonly used definition, since it is the least restrictive and the others are equiv-alent to it when X is Hausdorff. This equivalence is a consequence of the facts that compact subsets of Hausdorffspaces are closed, and closed subsets of compact spaces are compact.Condition (4) is used, for example, in Bourbaki.[1] In almost all applications, locally compact spaces are indeed alsoHausdorff. These locally compact Hausdorff (LCH) spaces are thus the spaces that this article is primarily concernedwith.

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48 CHAPTER 17. LOCALLY COMPACT SPACE

17.2 Examples and counterexamples

17.2.1 Compact Hausdorff spaces

Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in thearticle compact space. Here we mention only:

• the unit interval [0,1];

• the Cantor set;

• the Hilbert cube.

17.2.2 Locally compact Hausdorff spaces that are not compact

• The Euclidean spacesRn (and in particular the real lineR) are locally compact as a consequence of the Heine–Borel theorem.

• Topological manifolds share the local properties of Euclidean spaces and are therefore also all locally compact.This even includes nonparacompact manifolds such as the long line.

• All discrete spaces are locally compact and Hausdorff (they are just the zero-dimensional manifolds). Theseare compact only if they are finite.

• All open or closed subsets of a locally compact Hausdorff space are locally compact in the subspace topology.This provides several examples of locally compact subsets of Euclidean spaces, such as the unit disc (either theopen or closed version).

• The space Qp of p-adic numbers is locally compact, because it is homeomorphic to the Cantor set minus onepoint. Thus locally compact spaces are as useful in p-adic analysis as in classical analysis.

17.2.3 Hausdorff spaces that are not locally compact

Asmentioned in the following section, no Hausdorff space can possibly be locally compact if it is not also a Tychonoffspace; there are some examples of Hausdorff spaces that are not Tychonoff spaces in that article. But there are alsoexamples of Tychonoff spaces that fail to be locally compact, such as:

• the spaceQ of rational numbers (endowed with the topology fromR), since its compact subsets all have emptyinterior and therefore are not neighborhoods;

• the subspace (0,0) union (x,y) : x > 0 of R2, since the origin does not have a compact neighborhood;

• the lower limit topology or upper limit topology on the set R of real numbers (useful in the study of one-sidedlimits);

• any T0, hence Hausdorff, topological vector space that is infinite-dimensional, such as an infinite-dimensionalHilbert space.

The first two examples show that a subset of a locally compact space need not be locally compact, which contrastswith the open and closed subsets in the previous section. The last example contrasts with the Euclidean spaces inthe previous section; to be more specific, a Hausdorff topological vector space is locally compact if and only if it isfinite-dimensional (in which case it is a Euclidean space). This example also contrasts with the Hilbert cube as anexample of a compact space; there is no contradiction because the cube cannot be a neighbourhood of any point inHilbert space.

17.3. PROPERTIES 49

17.2.4 Non-Hausdorff examples

• The one-point compactification of the rational numbers Q is compact and therefore locally compact in senses(1) and (2) but it is not locally compact in sense (3).

• The particular point topology on any infinite set is locally compact in sense (3) but not in sense (2), because ithas no nonempty closed compact subspaces containing the particular point. The same holds for the real linewith the upper topology.

17.3 Properties

Every locally compact preregular space is, in fact, completely regular. It follows that every locally compact Hausdorffspace is a Tychonoff space. Since straight regularity is a more familiar condition than either preregularity (which isusually weaker) or complete regularity (which is usually stronger), locally compact preregular spaces are normallyreferred to in the mathematical literature as locally compact regular spaces. Similarly locally compact Tychonoffspaces are usually just referred to as locally compact Hausdorff spaces.Every locally compact Hausdorff space is a Baire space. That is, the conclusion of the Baire category theorem holds:the interior of every union of countably many nowhere dense subsets is empty.A subspace X of a locally compact Hausdorff space Y is locally compact if and only if X can be written as the set-theoretic difference of two closed subsets of Y. As a corollary, a dense subspace X of a locally compact Hausdorffspace Y is locally compact if and only if X is an open subset of Y. Furthermore, if a subspace X of any Hausdorffspace Y is locally compact, then X still must be the difference of two closed subsets of Y, although the converseneedn't hold in this case.Quotient spaces of locally compact Hausdorff spaces are compactly generated. Conversely, every compactly gener-ated Hausdorff space is a quotient of some locally compact Hausdorff space.For locally compact spaces local uniform convergence is the same as compact convergence.

17.3.1 The point at infinity

Since every locally compact Hausdorff space X is Tychonoff, it can be embedded in a compact Hausdorff space b(X)using the Stone–Čech compactification. But in fact, there is a simpler method available in the locally compact case;the one-point compactification will embed X in a compact Hausdorff space a(X) with just one extra point. (The one-point compactification can be applied to other spaces, but a(X) will be Hausdorff if and only if X is locally compactand Hausdorff.) The locally compact Hausdorff spaces can thus be characterised as the open subsets of compactHausdorff spaces.Intuitively, the extra point in a(X) can be thought of as a point at infinity. The point at infinity should be thought ofas lying outside every compact subset of X. Many intuitive notions about tendency towards infinity can be formulatedin locally compact Hausdorff spaces using this idea. For example, a continuous real or complex valued function fwith domain X is said to vanish at infinity if, given any positive number e, there is a compact subset K of X suchthat |f(x)| < e whenever the point x lies outside of K. This definition makes sense for any topological space X. IfX is locally compact and Hausdorff, such functions are precisely those extendable to a continuous function g on itsone-point compactification a(X) = X ∪ ∞ where g(∞) = 0.The set C0(X) of all continuous complex-valued functions that vanish at infinity is a C* algebra. In fact, everycommutative C* algebra is isomorphic to C0(X) for some unique (up to homeomorphism) locally compact Hausdorffspace X. More precisely, the categories of locally compact Hausdorff spaces and of commutative C* algebras aredual; this is shown using the Gelfand representation. Forming the one-point compactification a(X) of X correspondsunder this duality to adjoining an identity element to C0(X).

17.3.2 Locally compact groups

The notion of local compactness is important in the study of topological groups mainly because every Hausdorfflocally compact group G carries natural measures called the Haar measures which allow one to integrate measurablefunctions defined on G. The Lebesgue measure on the real line R is a special case of this.

50 CHAPTER 17. LOCALLY COMPACT SPACE

The Pontryagin dual of a topological abelian group A is locally compact if and only if A is locally compact. Moreprecisely, Pontryagin duality defines a self-duality of the category of locally compact abelian groups. The study oflocally compact abelian groups is the foundation of harmonic analysis, a field that has since spread to non-abelianlocally compact groups.

17.4 Notes[1] Bourbaki, Nicolas (1989). General Topology, Part I (reprint of the 1966 ed.). Berlin: Springer-Verlag. ISBN 3-540-

19374-X.

17.5 References• Kelley, John (1975). General Topology. Springer. ISBN 978-0387901251.

• Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 978-0131816299.

• 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. Addison-Wesley. ISBN 978-0486434797.

Chapter 18

Locally finite

The term locally finite has a number of different meanings in mathematics:

• Locally finite collection of sets in a topological space

• Locally finite group

• Locally finite measure

• Locally finite operator in linear algebra

• Locally finite poset

• Locally finite space, a topological space in which each point has a finite neighborhood

• Locally finite variety in the sense of universal algebra

51

Chapter 19

Locally finite collection

In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space.It is fundamental in the study of paracompactness and topological dimension.A collection of subsets of a topological spaceX is said to be locally finite, if each point in the space has a neighbourhoodthat intersects only finitely many of the sets in the collection.Note that the term locally finite has different meanings in other mathematical fields.

19.1 Examples and properties

A finite collection of subsets of a topological space is locally finite. Infinite collections can also be locally finite: forexample, the collection of all subsets of R of the form (n, n + 2) with integer n. A countable collection of subsetsneed not be locally finite, as shown by the collection of all subsets of R of the form (−n, n) with integer n.If a collection of sets is locally finite, the collection of all closures of these sets is also locally finite. The reasonfor this is that if an open set containing a point intersects the closure of a set, it necessarily intersects the set itself,hence a neighborhood can intersect at most the same number of closures (it may intersect fewer, since two distinct,indeed disjoint, sets can have the same closure). The converse, however, can fail if the closures of the sets are notdistinct. For example, in the finite complement topology on R the collection of all open sets is not locally finite, butthe collection of all closures of these sets is locally finite (since the only closures are R and the empty set).

19.1.1 Compact spaces

No infinite collection of a compact space can be locally finite. Indeed, let Ga be an infinite family of subsets ofa space and suppose this collection is locally finite. For each point x of this space, choose a neighbourhood Ux thatintersects the collection Ga at only finitely many values of a. Clearly:

Ux for each x in X (the union over all x) is an open covering in X

and hence has a finite subcover, Ua1 ∪ ...... ∪ Uan. Since each Uai intersects Ga for only finitely many values ofa, the union of all such Uai intersects the collection Ga for only finitely many values of a. It follows that X (thewhole space!) intersects the collection Ga at only finitely many values of a, contradicting the infinite cardinality ofthe collection Ga.A topological space in which every open cover admits a locally finite open refinement is called paracompact. Everylocally finite collection of subsets of a topological space X is also point-finite. A topological space in which everyopen cover admits a point-finite open refinement is called metacompact.

19.1.2 Second countable spaces

No uncountable cover of a Lindelöf space can be locally finite, by essentially the same argument as in the case ofcompact spaces. In particular, no uncountable cover of a second-countable space is locally finite.

52

19.2. CLOSED SETS 53

19.2 Closed sets

It is clear from the definition of a topology that a finite union of closed sets is closed. One can readily give an exampleof an infinite union of closed sets that is not closed. However, if we consider a locally finite collection of closed sets,the union is closed. To see this we note that if x is a point outside the union of this locally finite collection of closedsets, we merely choose a neighbourhoodV of x that intersects this collection at only finitely many of these sets. Definea bijective map from the collection of sets that V intersects to 1, ..., k thus giving an index to each of these sets.Then for each set, choose an open set Ui containing x that doesn't intersect it. The intersection of all such Ui for 1 ≤i ≤ k intersected with V, is a neighbourhood of x that does not intersect the union of this collection of closed sets.

19.3 Countably locally finite collections

A collection in a space is countably locally finite (or σ-locally finite) if it is the union of a countable family of locallyfinite collections of subsets of X. Countable local finiteness is a key hypothesis in the Nagata–Smirnov metrizationtheorem, which states that a topological space is metrizable if and only if it is regular and has a countably locally finitebasis.

19.4 References• James R. Munkres (2000), Topology (2nd ed.), Prentice Hall, ISBN 0-13-181629-2

Chapter 20

Locally finite space

In the mathematical field of topology, a locally finite space is a topological space in which every point has a finiteneighborhood.A locally finite space is Alexandrov.A T1 space is locally finite if and only if it is discrete.

20.1 References• Nakaoka, Fumie; Oda, Nobuyuki (2001), “Some applications of minimal open sets”, International Journal ofMathematics and Mathematical Sciences 29 (8): 471–476, doi:10.1155/S0161171201006482

54

Chapter 21

Mesocompact space

In mathematics, in the field of general topology, a topological space is said to bemesocompact if every open cover hasa compact-finite open refinement.[1] That is, given any open cover, we can find an open refinement with the propertythat every compact set meets only finitely many members of the refinement.[2]

The following facts are true about mesocompactness:

• Every compact space, and more generally every paracompact space is mesocompact. This follows from thefact that any locally finite cover is automatically compact-finite.

• Every mesocompact space is metacompact, and hence also orthocompact. This follows from the fact that pointsare compact, and hence any compact-finite cover is automatically point finite.

21.1 Notes[1] Hart, Nagata & Vaughan, p200

[2] Pearl, p23

21.2 References• K.P. Hart; J. Nagata; J.E. Vaughan, eds. (2004), Encyclopedia of General Topology, Elsevier, ISBN 0-444-50355-2

• Pearl, Elliott, ed. (2007), Open Problems in Topology II, Elsevier, ISBN 0-444-52208-5

55

Chapter 22

Metacompact space

In mathematics, in the field of general topology, a topological space is said to be metacompact if every open coverhas a point finite open refinement. That is, given any open cover of the topological space, there is a refinement whichis again an open cover with the property that every point is contained only in finitely many sets of the refining cover.A space is countably metacompact if every countable open cover has a point finite open refinement.

22.1 Properties

The following can be said about metacompactness in relation to other properties of topological spaces:

• Every paracompact space is metacompact. This implies that every compact space is metacompact, and everymetric space is metacompact. The converse does not hold: a counter-example is the Dieudonné plank.

• Every metacompact space is orthocompact.

• Every metacompact normal space is a shrinking space

• The product of a compact space and a metacompact space is metacompact. This follows from the tube lemma.

• An easy example of a non-metacompact space (but a countably metacompact space) is the Moore plane.

• In order for a Tychonoff space X to be compact it is necessary and sufficient that X be metacompact andpseudocompact (see Watson).

22.2 Covering dimension

A topological space X is said to be of covering dimension n if every open cover of X has a point finite open refinementsuch that no point of X is included in more than n + 1 sets in the refinement and if n is the minimum value for whichthis is true. If no such minimal n exists, the space is said to be of infinite covering dimension.

22.3 See also

• Compact space

• Paracompact space

• Normal space

• Realcompact space

• Pseudocompact space

56

22.4. REFERENCES 57

• Mesocompact space

• Tychonoff space

• Glossary of topology

22.4 References• Watson, W. Stephen (1981). “Pseudocompact metacompact spaces are compact”. Proc. Amer. Math. Soc.81: 151–152. doi:10.1090/s0002-9939-1981-0589159-1.

• 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. P.23.

Chapter 23

Orthocompact space

In mathematics, in the field of general topology, a topological space is said to be orthocompact if every open coverhas an interior preserving open refinement. That is, given an open cover of the topological space, there is a refinementwhich is also an open cover, with the further property that at any point, the intersection of all open sets in the refinementcontaining that point, is also open.If the number of open sets containing the point is finite, then their intersection is clearly open. That is, every pointfinite open cover is interior preserving. Hence, we have the following: every metacompact space, and in particular,every paracompact space, is orthocompact.Useful theorems:

• Orthocompactness is a topological invariant; that is, it is preserved by homeomorphisms.

• Every closed subspace of an orthocompact space is orthocompact.

• A topological space X is orthocompact if and only if every open cover of X by basic open subsets of X has aninterior-preserving refinement that is an open cover of X.

• The product X × [0,1] of the closed unit interval with an orthocompact space X is orthocompact if and only ifX is countably metacompact. (B.M. Scott) [1]

• Every orthocompact space is countably orthocompact.

• Every countably orthocompact Lindelöf space is orthocompact.

23.1 References[1] B.M. Scott, Towards a product theory for orthocompactness, “Studies in Topology”, N.M. Stavrakas and K.R. Allen, eds

(1975), 517–537.

• P. Fletcher, W.F. Lindgren, Quasi-uniform Spaces, Marcel Dekker, 1982, ISBN 0-8247-1839-9. Chap.V.

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Chapter 24

Paracompact space

In mathematics, a paracompact space is a topological space in which every open cover has an open refinement thatis locally finite. These spaces were introduced by Dieudonné (1944). Every compact space is paracompact. Everyparacompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions ofunity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces arealways closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact spaceis called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact.Tychonoff’s theorem (which states that the product of any collection of compact topological spaces is compact) doesnot generalize to paracompact spaces in that the product of paracompact spaces need not be paracompact. However,the product of a paracompact space and a compact space is always paracompact.Every metric space is paracompact. A topological space is metrizable if and only if it is a paracompact and locallymetrizable Hausdorff space.

24.1 Paracompactness

A cover of a set X is a collection of subsets of X whose union contains X. In symbols, if U = Uα : α in A is anindexed family of subsets of X, then U is a cover of X if

X ⊆∪α∈A

Uα.

A cover of a topological space X is open if all its members are open sets. A refinement of a cover of a space X is anew cover of the same space such that every set in the new cover is a subset of some set in the old cover. In symbols,the cover V = Vᵦ : β in B is a refinement of the cover U = Uα : α in A if and only if, for any Vᵦ in V, thereexists some Uα in U such that Vᵦ⊆Uα.An open cover of a space X is locally finite if every point of the space has a neighborhood that intersects only finitelymany sets in the cover. In symbols, U = Uα : α in A is locally finite if and only if, for any x in X, there exists someneighbourhood V(x) of x such that the set

α ∈ A : Uα ∩ V (x) = ∅

is finite. A topological spaceX is now said to be paracompact if every open cover has a locally finite open refinement.

24.2 Examples• Every compact space is paracompact.

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60 CHAPTER 24. PARACOMPACT SPACE

• Every regular Lindelöf space is paracompact. In particular, every locally compact Hausdorff second-countablespace is paracompact.

• The Sorgenfrey line is paracompact, even though it is neither compact, locally compact, second countable, normetrizable.

• Every CW complex is paracompact [1]

• (Theorem of A. H. Stone) Every metric space is paracompact.[2] Early proofs were somewhat involved, butan elementary one was found by M. E. Rudin.[3] Existing proofs of this require the axiom of choice for thenon-separable case. It has been shown that neither ZF theory nor ZF theory with the axiom of dependentchoice is sufficient.[4]

Some examples of spaces that are not paracompact include:

• The most famous counterexample is the long line, which is a nonparacompact topological manifold. (The longline is locally compact, but not second countable.)

• Another counterexample is a product of uncountably many copies of an infinite discrete space. Any infiniteset carrying the particular point topology is not paracompact; in fact it is not even metacompact.

• The Prüfer manifold is a non-paracompact surface.

• The bagpipe theorem shows that there are 2ℵ1 isomorphism classes of non-paracompact surfaces.

24.3 Properties

Paracompactness is weakly hereditary, i.e. every closed subspace of a paracompact space is paracompact. This canbe extended to F-sigma subspaces as well.

• A regular space is paracompact if every open cover admits a locally finite refinement. (Here, the refinement isnot required to be open.) In particular, every regular Lindelof space is paracompact.

• (Smirnov metrization theorem) A topological space is metrizable if and only if it is paracompact, Hausdorff,and locally metrizable.

• Michael selection theorem states that lower semicontinuousmultifunctions fromX into nonempty closed convexsubsets of Banach spaces admit continuous selection iff X is paracompact.

Although a product of paracompact spaces need not be paracompact, the following are true:

• The product of a paracompact space and a compact space is paracompact.

• The product of a metacompact space and a compact space is metacompact.

Both these results can be proved by the tube lemma which is used in the proof that a product of finitely many compactspaces is compact.

24.4 Paracompact Hausdorff Spaces

Paracompact spaces are sometimes required to also be Hausdorff to extend their properties.

• (Theorem of Jean Dieudonné) Every paracompact Hausdorff space is normal.

• Every paracompact Hausdorff space is a shrinking space, that is, every open cover of a paracompact Hausdorffspace has a shrinking: another open cover indexed by the same set such that the closure of every set in the newcover lies inside the corresponding set in the old cover.

• On paracompact Hausdorff spaces, sheaf cohomology and Čech cohomology are equal.[5]

24.4. PARACOMPACT HAUSDORFF SPACES 61

24.4.1 Partitions of unity

The most important feature of paracompact Hausdorff spaces is that they are normal and admit partitions of unitysubordinate to any open cover. This means the following: if X is a paracompact Hausdorff space with a given opencover, then there exists a collection of continuous functions on X with values in the unit interval [0, 1] such that:

• for every function f: X → R from the collection, there is an open set U from the cover such that the supportof f is contained in U;

• for every point x in X, there is a neighborhood V of x such that all but finitely many of the functions in thecollection are identically 0 in V and the sum of the nonzero functions is identically 1 in V.

In fact, a T1 space is Hausdorff and paracompact if and only if it admits partitions of unity subordinate to any opencover (see below). This property is sometimes used to define paracompact spaces (at least in the Hausdorff case).Partitions of unity are useful because they often allow one to extend local constructions to the whole space. Forinstance, the integral of differential forms on paracompact manifolds is first defined locally (where the manifold lookslike Euclidean space and the integral is well known), and this definition is then extended to the whole space via apartition of unity.

Proof that paracompact Hausdorff spaces admit partitions of unity

A Hausdorff space X is paracompact if and only if it every open cover admits a subordinate partition of unity. Theif direction is straightforward. Now for the only if direction, we do this in a few stages.

Lemma 1: If O is a locally finite open cover, then there exists open setsWU for each U ∈ O , suchthat each WU ⊆ U and WU : U ∈ O is a locally finite refinement.

Lemma 2: IfO is a locally finite open cover, then there are continuous functions fU : X → [0, 1] suchthat supp fU ⊆ U and such that f :=

∑U∈O fU is a continuous function which is always non-zero

and finite.

Theorem: In a paracompact Hausdorff spaceX , ifO is an open cover, then there exists a partition ofunity subordinate to it.

Proof (Lemma 1): Let V be the collection of open sets meeting only finitely many sets in O , andwhose closure is contained in a set in O . One can check as an exercise that this provides an openrefinement, since paracompact Hausdorff spaces are regular, and since O is locally finite. Now replaceV by a locally finite open refinement. One can easily check that each set in this refinement has the sameproperty as that which characterised the original cover.

Now we defineWU =∪A ∈ V : A ⊆ U . We have that each WU ⊆ U ; for otherwise: suppose

there is x ∈ WU \ U . We will show that there is closed set C ⊃ WU such that x /∈ C (this meanssimply x /∈ WU by definition of closure). Since we chose V to be locally finite there is neighbourhoodV [x] of x such that only finitely many sets U1, ..., Un ∈ A ∈ V : A ⊆ U have non-empty intersectionwith V [x] . We take their closures U1, ..., Un and then V := V [x] \ ∪Ui is an open set (since sum isfinite) such that V ∩WU = ∅ . Moreover x ∈ V , because ∀i = 1, ..., n we have Ui ⊆ U and weknow that x /∈ U . Then C := X \ V is closed set without x which conatins WU . So x /∈ WU andwe've reached contradiction. And it easy to see that WU : U ∈ O is an open refinement of O .

Finally, to verify that this cover is locally finite, fix x ∈ X and letN be a neighbourhood of x . We knowthat for each U we haveWU ⊆ U . Since O is locally finite there are only finitely many sets U1, ..., Uk

having non-empty intersection withN . Then only setsWU1 , ...,WUkhave non-empty intersection with

N , because for every other U ′ we have N ∩WU ′ ⊆ N ∩ U ′ = ∅

62 CHAPTER 24. PARACOMPACT SPACE

Proof (Lemma 2): Applying Lemma 1, let fU : X → [0, 1] be coninuous maps with fU WU = 1and supp fU ⊆ U (by Urysohn’s lemma for disjoint closed sets in normal spaces, which a paracompactHausdorff space is). Note by the support of a function, we here mean the points not mapping to zero(and not the closure of this set). To show that f =

∑U∈O fU is always finite and non-zero, take x ∈ X

, and let N a neighbourhood of x meeting only finitely many sets in O ; thus x belongs to only finitelymany sets in O ; thus fU (x) = 0 for all but finitely many U ; moreover x ∈ WU for some U ,thus fU (x) = 1 ; so f(x) is finite and ≥ 1 . To establish continuity, take x,N as before, and letS = U ∈ O : N meets U , which is finite; then f N =

∑U∈S fU N , which is a continuous

function; hence the preimage under f of a neighbourhood of f(x) will be a neighbourhood of x .

Proof (Theorem): TakeO∗ a locally finite subcover of the refinement cover: V open : (∃U ∈ O)V ⊆U . Applying Lemma 2, we obtain continuous functions fW : X → [0, 1] with supp fW ⊆W (thusthe usual closed version of the support is contained in some U ∈ O , for eachW ∈ O∗ ; for which theirsum constitutes a continuous function which is always finite non-zero (hence 1/f is continuous positive,finite-valued). So replacing each fW by fW /f , we have now — all things remaining the same — thattheir sum is everywhere 1 . Finally for x ∈ X , lettingN be a neighbourhood of x meeting only finitelymany sets in O∗ , we have fW N = 0 for all but finitely manyW ∈ O∗ since each supp fW ⊆W. Thus we have a partition of unity subordinate to the original open cover.

24.5 Relationship with compactness

There is a similarity between the definitions of compactness and paracompactness: For paracompactness, “subcover”is replaced by “open refinement” and “finite” by is replaced by “locally finite”. Both of these changes are significant:if we take the definition of paracompact and change “open refinement” back to “subcover”, or “locally finite” back to“finite”, we end up with the compact spaces in both cases.Paracompactness has little to do with the notion of compactness, but rather more to do with breaking up topologicalspace entities into manageable pieces.

24.5.1 Comparison of properties with compactness

Paracompactness is similar to compactness in the following respects:

• Every closed subset of a paracompact space is paracompact.

• Every paracompact Hausdorff space is normal.

It is different in these respects:

• A paracompact subset of a Hausdorff space need not be closed. In fact, for metric spaces, all subsets areparacompact.

• A product of paracompact spaces need not be paracompact. The square of the real line R in the lower limittopology is a classical example for this.

24.6 Variations

There are several variations of the notion of paracompactness. To define them, we first need to extend the list ofterms above:A topological space is:

• metacompact if every open cover has an open pointwise finite refinement.

• orthocompact if every open cover has an open refinement such that the intersection of all the open sets aboutany point in this refinement is open.

24.7. SEE ALSO 63

• fully normal if every open cover has an open star refinement, and fully T4 if it is fully normal and T1 (seeseparation axioms).

The adverb "countably" can be added to any of the adjectives “paracompact”, “metacompact”, and “fully normal” tomake the requirement apply only to countable open covers.Every paracompact space is metacompact, and every metacompact space is orthocompact.

24.6.1 Definition of relevant terms for the variations

• Given a cover and a point, the star of the point in the cover is the union of all the sets in the cover that containthe point. In symbols, the star of x in U = Uα : α in A is

U∗(x) :=∪

Uα∋x

Uα.

The notation for the star is not standardised in the literature, and this is just one possibility.

• A star refinement of a cover of a space X is a new cover of the same space such that, given any point in thespace, the star of the point in the new cover is a subset of some set in the old cover. In symbols, V is a starrefinement of U = Uα : α in A if and only if, for any x in X, there exists a Uα in U, such that V*(x) iscontained in Uα.

• A cover of a space X is pointwise finite if every point of the space belongs to only finitely many sets in thecover. In symbols, U is pointwise finite if and only if, for any x in X, the set

α ∈ A : x ∈ Uα

is finite.

As the name implies, a fully normal space is normal. Every fully T4 space is paracompact. In fact, for Hausdorffspaces, paracompactness and full normality are equivalent. Thus, a fully T4 space is the same thing as a paracompactHausdorff space.As an historical note: fully normal spaces were defined before paracompact spaces. The proof that all metrizablespaces are fully normal is easy. When it was proved by A.H. Stone that for Hausdorff spaces fully normal andparacompact are equivalent, he implicitly proved that all metrizable spaces are paracompact. Later M.E. Rudin gavea direct proof of the latter fact.

24.7 See also• a-paracompact space

• Paranormal space

24.8 Notes[1] Hatcher, Allen, Vector bundles and K-theory, preliminary version available on the author’s homepage

[2] Stone, A. H. Paracompactness and product spaces. Bull. Amer. Math. Soc. 54 (1948), 977-982

[3] Rudin, Mary Ellen. A new proof that metric spaces are paracompact. Proceedings of the American Mathematical Society,Vol. 20, No. 2. (Feb., 1969), p. 603.

[4] C. Good, I. J. Tree, and W. S. Watson. On Stone’s Theorem and the Axiom of Choice. Proceedings of the AmericanMathematical Society, Vol. 126, No. 4. (April, 1998), pp. 1211–1218.

[5] Brylinski, Jean-Luc (2007), Loop Spaces, Characteristic Classes and Geometric Quantization, Progress in Mathematics 107,Springer, p. 32, ISBN 9780817647308.

64 CHAPTER 24. PARACOMPACT SPACE

24.9 References• Dieudonné, Jean (1944), “Une généralisation des espaces compacts”, Journal de Mathématiques Pures et Ap-pliquées, Neuvième Série 23: 65–76, ISSN 0021-7824, MR 0013297

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology (2 ed), Springer Verlag, 1978,ISBN 3-540-90312-7. P.23.

• Willard, Stephen (1970). General Topology. Reading, Massachusetts: Addison-Wesley. ISBN 0-486-43479-6.

• Mathew, Akhil. “Topology/Paracompactness”.

24.10 External links• Hazewinkel, Michiel, ed. (2001), “Paracompact space”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 25

Pseudocompact space

In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under anycontinuous function to R is bounded.

25.1 Properties related to pseudocompactness

• In order that a Tychonoff space X be pseudocompact it is necessary and sufficient that every locally finitecollection of non-empty open sets of X be finite. A series of equivalent conditions was given by Kerstan andYan-Min and other authors (see the references).

• Every countably compact space is pseudocompact. For normal Hausdorff spaces the converse is true.

• As a consequence of the above result, every sequentially compact space is pseudocompact. The converse istrue for metric spaces. As sequential compactness is an equivalent condition to compactness for metric spacesthis implies that compactness is an equivalent condition to pseudocompactness for metric spaces also.

• The weaker result that every compact space is pseudocompact is easily proved: the image of a compact spaceunder any continuous function is compact, and the Heine–Borel theorem tells us that the compact subsets of Rare precisely the closed and bounded subsets.

• If Y is the continuous image of pseudocompactX, then Y is pseudocompact. Note that for continuous functionsg : X → Y and h : Y → R, the composition of g and h, called f, is a continuous function from X to the realnumbers. Therefore, f is bounded, and Y is pseudocompact.

• Let X be an infinite set given the particular point topology. Then X is neither compact, sequentially compact,countably compact, paracompact nor metacompact. However, since X is hyperconnected, it is pseudocompact.This shows that pseudocompactness doesn't imply any other (known) form of compactness.

• In order that a Hausdorff space X be compact it is necessary and sufficient that X be pseudocompact andrealcompact (see Engelking, p. 153).

• In order that a Tychonoff space X be compact it is necessary and sufficient that X be pseudocompact andmetacompact (see Watson).

25.2 See also

• Compact space

• Paracompact space

65

66 CHAPTER 25. PSEUDOCOMPACT SPACE

• Normal space

• Realcompact space

• Metacompact space

• Tychonoff space

25.3 References• Kerstan, Johannes (1957), “Zur Charakterisierung der pseudokompakten Räume”,Mathematische Nachrichten16 (5–6): 289–293, doi:10.1002/mana.19570160505.

• W. Stephen, Watson (1981), “Pseudocompact metacompact spaces are compact”, Proc. Amer. Math. Soc. 81:151–152, doi:10.1090/s0002-9939-1981-0589159-1.

• Yan-Min, Wang (1988), “New characterisations of pseudocompact spaces”, Bull. Austral. Math. Soc. 38 (2):293–298, doi:10.1017/S0004972700027568.

• Engelking, Ryszard (1968), Outline of General Topology, translated from Polish, Amsterdam: North-Holland.

• Willard, Stephen (1970), General Topology, Reading, Mass.: Addison-Wesley.

• M.I. Voitsekhovskii (2001), “Pseudo-compact space”, in Hazewinkel, Michiel, Encyclopedia of Mathematics,Springer, ISBN 978-1-55608-010-4.

• Pseudocompact space at PlanetMath.org. .

Chapter 26

Realcompact space

In mathematics, in the field of topology, a topological space is said to be realcompact if it is completely regularHausdorff and every point of its Stone–Čech compactification is real (meaning that the quotient field at that pointof the ring of real functions is the reals). Realcompact spaces have also been called Q-spaces, saturated spaces,functionally complete spaces, real-complete spaces, replete spaces and Hewitt-Nachbin spaces (named afterEdwin Hewitt and Leopoldo Nachbin). Realcompact spaces were introduced by Hewitt (1948).

26.1 Properties

• A space is realcompact if and only if it can be embedded homeomorphically as a closed subset in some (notnecessarily finite) Cartesian power of the reals, with the product topology. Moreover, a (Hausdorff) space isrealcompact if and only if it has the uniform topology and is complete for the uniform structure generated bythe continuous real-valued functions (Gillman, Jerison, p. 226).

• For example Lindelöf spaces are realcompact; in particular all subsets of Rn are realcompact.

• The (Hewitt) realcompactification υX of a topological space X consists of the real points of its Stone–Čechcompactification βX. A topological space X is realcompact if and only if it coincides with its Hewitt realcom-pactification.

• Write C(X) for the ring of continuous functions on a topological spaceX. If Y is a real compact space, then ringhomomorphisms from C(Y) to C(X) correspond to continuous maps from X to Y. In particular the category ofrealcompact spaces is dual to the category of rings of the form C(X).

• In order that a Hausdorff space X is compact it is necessary and sufficient that X is realcompact and pseu-docompact (see Engelking, p. 153).

26.2 See also

• Compact space

• Paracompact space

• Normal space

• Pseudocompact space

• Tychonoff space

67

68 CHAPTER 26. REALCOMPACT SPACE

26.3 References• Gillman, Leonard; Jerison, Meyer, “Rings of continuous functions”. Reprint of the 1960 edition. GraduateTexts in Mathematics, No. 43. Springer-Verlag, New York-Heidelberg, 1976. xiii+300 pp.

• Hewitt, Edwin (1948), “Rings of real-valued continuous functions. I”, Transactions of the American Mathe-matical Society 64: 45–99, ISSN 0002-9947, JSTOR 1990558, MR 0026239.

• Engelking, Ryszard (1968). Outline of General Topology. translated from Polish. Amsterdam: North-HollandPubl. Co..

• Willard, Stephen (1970), General Topology, Reading, Mass.: Addison-Wesley.

Chapter 27

Relatively compact subspace

In mathematics, a relatively compact subspace (or relatively compact subset) Y of a topological spaceX is a subsetwhose closure is compact.Since closed subsets of a compact space are compact, every subset of a compact space is relatively compact. In thecase of a metric topology, or more generally when sequences may be used to test for compactness, the criterion forrelative compactness becomes that any sequence in Y has a subsequence convergent in X. Such a subset may also becalled relatively bounded, or pre-compact, although the latter term is also used for a totally bounded subset. (Theseare equivalent in a complete space.)Some major theorems characterise relatively compact subsets, in particular in function spaces. An example is theArzelà–Ascoli theorem. Other cases of interest relate to uniform integrability, and the concept of normal familyin complex analysis. Mahler’s compactness theorem in the geometry of numbers characterises relatively compactsubsets in certain non-compact homogeneous spaces (specifically spaces of lattices).The definition of an almost periodic function F at a conceptual level has to do with the translates of F being a relativelycompact set. This needs to be made precise in terms of the topology used, in a particular theory.As a counterexample take any neighbourhood of the particular point of an infinite particular point space. The neigh-bourhood itself may be compact but is not relatively compact because its closure is the whole non-compact space.

27.1 See also• Compactly embedded

27.2 References• page 12 of V. Khatskevich, D.Shoikhet, Differentiable Operators and Nonlinear Equations, Birkhäuser VerlagAG, Basel, 1993, 270 pp. at google books

69

Chapter 28

Sequentially compact space

In mathematics, a topological space is sequentially compact if every infinite sequence has a convergent subsequence.For general topological spaces, the notions of compactness and sequential compactness are not equivalent; they are,however, equivalent for metric spaces. A metric space X is sequentially compact if every sequence has a convergentsubsequence which converges to a point in X.

28.1 Examples and properties

The space of all real numbers with the standard topology is not sequentially compact; the sequence (sn = n) for allnatural numbers n is a sequence that has no convergent subsequence.If a space is a metric space, then it is sequentially compact if and only if it is compact.[1] However in general thereexist sequentially compact spaces that are not compact (such as the first uncountable ordinal with the order topology),and compact spaces that are not sequentially compact (such as the product of 2ℵ0 = c copies of the closed unitinterval).[2]

28.2 Related notions

• A topological space X is said to be limit point compact if every infinite subset of X has a limit point in X.

• A topological space is countably compact if every countable open cover has a finite subcover.

In a metric space, the notions of sequential compactness, limit point compactness, countable compactness andcompactness are equivalent.In a sequential space sequential compactness is equivalent to countable compactness.[3]

There is also a notion of a one-point sequential compactification -- the idea is that the non convergent sequencesshould all converge to the extra point. See [4]

28.3 See also

• Bolzano–Weierstrass theorem

28.4 Notes[1] Willard, 17G, p. 125.

[2] Steen and Seebach, Example 105, pp. 125—126.

70

28.5. REFERENCES 71

[3] Engelking, General Topology, Theorem 3.10.31K.P. Hart, Jun-iti Nagata, J.E. Vaughan (editors), Encyclopedia of General Topology, Chapter d3 (by P. Simon)

[4] Brown, Ronald, “Sequentially proper maps and a sequential compactification”, J. London Math Soc. (2) 7 (1973) 515-522.

28.5 References• Munkres, James (1999). Topology (2nd edition ed.). Prentice Hall. ISBN 0-13-181629-2.

• Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970).ISBN 0-03-079485-4.

• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

Chapter 29

Strictly singular operator

In functional analysis, a branch of mathematics, a strictly singular operator is a bounded linear operator L from aBanach space X to another Banach space Y, such that it is not an isomorphism, and fails to be an isomorphism on anyinfinite dimensional subspace of X. Any compact operator is strictly singular, but not vice versa.[1][2]

Every bounded linear map T : lp → lq , for 1 ≤ q, p <∞ , p = q , is strictly singular. Here, lp and lq are sequencespaces. Similarly, every bounded linear map T : c0 → lp and T : lp → c0 , for 1 ≤ p < ∞ , is strictly singular.Here c0 is the Banach space of sequences converging to zero. This is a corollary of Pitt’s theorem, which states thatsuch T, for q < p, are compact.

29.1 References[1] N.L. Carothers, A Short Course on Banach Space Theory, (2005) London Mathematical Society Student Texts 64, Cam-

bridge University Press.

[2] C. J. Read, Strictly singular operators and the invariant subspace problem, Studia Mathematica 132 (3) (1999), 203-226.fulltext

72

Chapter 30

Subset

“Superset” redirects here. For other uses, see Superset (disambiguation).In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is

AB

Euler diagram showingA is a proper subset of B and conversely B is a proper superset of A

73

74 CHAPTER 30. SUBSET

“contained” inside B, that is, all elements of A are also elements of B. A and B may coincide. The relationship of oneset being a subset of another is called inclusion or sometimes containment.The subset relation defines a partial order on sets.The algebra of subsets forms a Boolean algebra in which the subset relation is called inclusion.

30.1 Definitions

If A and B are sets and every element of A is also an element of B, then:

• A is a subset of (or is included in) B, denoted by A ⊆ B ,or equivalently

• B is a superset of (or includes) A, denoted by B ⊇ A.

If A is a subset of B, but A is not equal to B (i.e. there exists at least one element of B which is not an element of A),then

• A is also a proper (or strict) subset of B; this is written as A ⊊ B.

or equivalently

• B is a proper superset of A; this is written as B ⊋ A.

For any set S, the inclusion relation ⊆ is a partial order on the set P(S) of all subsets of S (the power set of S).When quantified, A ⊆ B is represented as: ∀xx∈A → x∈B.[1]

30.2 ⊂ and ⊃ symbols

Some authors use the symbols ⊂ and ⊃ to indicate subset and superset respectively; that is, with the same meaningand instead of the symbols, ⊆ and ⊇.[2] So for example, for these authors, it is true of every set A that A ⊂ A.Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively, instead of ⊊ and⊋.[3] This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may ormay not equal y, but if x < y, then x may not equal y, and is less than y. Similarly, using the convention that ⊂ isproper subset, if A ⊆ B, then A may or may not equal B, but if A ⊂ B, then A definitely does not equal B.

30.3 Examples

• The set A = 1, 2 is a proper subset of B = 1, 2, 3, thus both expressions A ⊆ B and A ⊊ B are true.

• The set D = 1, 2, 3 is a subset of E = 1, 2, 3, thus D ⊆ E is true, and D ⊊ E is not true (false).

• Any set is a subset of itself, but not a proper subset. (X ⊆ X is true, and X ⊊ X is false for any set X.)

• The empty set , denoted by ∅, is also a subset of any given set X. It is also always a proper subset of any setexcept itself.

• The set x: x is a prime number greater than 10 is a proper subset of x: x is an odd number greater than 10

• The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a linesegment is a proper subset of the set of points in a line. These are two examples in which both the subset andthe whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is,the number of elements, of a finite set) as the whole; such cases can run counter to one’s initial intuition.

• The set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinitebut the latter set has a larger cardinality (or power) than the former set.

30.4. OTHER PROPERTIES OF INCLUSION 75

polygonsregular

polygons

The regular polygons form a subset of the polygons

Another example in an Euler diagram:

• A is a proper subset of B

• C is a subset but no proper subset of B

30.4 Other properties of inclusion

Inclusion is the canonical partial order in the sense that every partially ordered set (X, ⪯ ) is isomorphic to somecollection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identifiedwith the set [n] of all ordinals less than or equal to n, then a ≤ b if and only if [a] ⊆ [b].For the power set P(S) of a set S, the inclusion partial order is (up to an order isomorphism) the Cartesian product ofk = |S| (the cardinality of S) copies of the partial order on 0,1 for which 0 < 1. This can be illustrated by enumeratingS = s1, s2, …, sk and associating with each subset T ⊆ S (which is to say with each element of 2S) the k-tuple from0,1k of which the ith coordinate is 1 if and only if si is a member of T.

30.5 See also• Containment order

30.6 References[1] Rosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119. ISBN

978-0-07-338309-5.

[2] Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1,MR 924157

76 CHAPTER 30. SUBSET

CB

A

A B and B C imply A C

[3] Subsets and Proper Subsets (PDF), retrieved 2012-09-07

• Jech, Thomas (2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2.

30.7 External links• Weisstein, Eric W., “Subset”, MathWorld.

Chapter 31

Supercompact space

In mathematics, in the field of topology, a topological space is called supercompact if there is a subbasis such thatevery open cover of the topological space from elements of the subbasis has a subcover with at most two subbasiselements. Supercompactness and the related notion of superextension was introduced by J. de Groot in 1967.

31.1 Examples

By the Alexander subbase theorem, every supercompact space is compact. Conversely, many (but not all) compactspaces are supercompact. The following are examples of supercompact spaces:

• Compact linearly ordered spaces with the order topology and all continuous images of such spaces (Bula et al.1992)

• Compact metrizable spaces (due originally to M. Strok and A. Szymański 1975, see also Mills 1979)

• A product of supercompact spaces is supercompact (like a similar statement about compactness, Tychonoff’stheorem, it is equivalent to the axiom of choice, Banaschewski 1993)

31.2 Some Properties

Some compact Hausdorff spaces are not supercompact; such an example is given by the Stone–Čech compactificationof the natural numbers (with the discrete topology) (Bell 1978).A continuous image of a supercompact space need not be supercompact (Verbeek 1972, Mills—van Mill 1979).In a supercompact space (or any continuous image of one), the cluster point of any countable subset is the limit of anontrivial convergent sequence. (Yang 1994)

31.3 References

• B. Banaschewski, “Supercompactness, products and the axiom of choice.” Kyungpook Math. J. 33 (1993), no.1, 111—114.

• Bula, W.; Nikiel, J.; Tuncali, H. M.; Tymchatyn, E. D. “Continuous images of ordered compacta are regularsupercompact.” Proceedings of the Tsukuba Topology Symposium (Tsukuba, 1990). Topology Appl. 45(1992), no. 3, 203—221.

• Murray G. Bell. “Not all compact Hausdorff spaces are supercompact.” General Topology and Appl. 8 (1978),no. 2, 151—155.

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78 CHAPTER 31. SUPERCOMPACT SPACE

• J. de Groot, “Supercompactness and superextensions.” Contributions to extension theory of topological struc-tures. Proceedings of the Symposium held in Berlin, August 14—19, 1967. Edited by J. Flachsmeyer, H.Poppe and F. Terpe. VEB Deutscher Verlag der Wissenschaften, Berlin 1969 279 pp.

• Engelking, R (1977), General topology, Taylor & Francis, ISBN 978-0-8002-0209-5.

• Malykhin, VI; Ponomarev, VI (1977), “General topology (set-theoretic trend)", Journal of Mathematical Sci-ences (New York: Springer) 7 (4): 587–629, doi:10.1007/BF01084982

• Mills, Charles F. (1979), “A simpler proof that compact metric spaces are supercompact”, Proceedings of theAmerican Mathematical Society (Proceedings of the American Mathematical Society, Vol. 73, No. 3) 73 (3):388–390, doi:10.2307/2042369, JSTOR 2042369, MR 518526

• Mills, Charles F.; van Mill, Jan, “A nonsupercompact continuous image of a supercompact space.” Houston J.Math. 5 (1979), no. 2, 241—247.

• Mysior, Adam (1992), “Universal compact T1-spaces”, Canadian Mathematical Bulletin (Canadian Mathemat-ical Society) 35 (2): 261–266, doi:10.4153/CMB-1992-037-1.

• J. van Mill, Supercompactness and Wallman spaces. Mathematical Centre Tracts, No. 85. MathematischCentrum, Amsterdam, 1977. iv+238 pp. ISBN 90-6196-151-3

• M. Strok and A. Szymanski, "Compact metric spaces have binary bases. " Fund. Math. 89 (1975), no. 1,81—91.

• A. Verbeek, Superextensions of topological spaces. Mathematical Centre Tracts, No. 41. Mathematisch Cen-trum, Amsterdam, 1972. iv+155 pp.

• Yang, Zhong Qiang (1994), “All cluster points of countable sets in supercompact spaces are the limits ofnontrivial sequences”, Proceedings of the American Mathematical Society (Proceedings of the American Math-ematical Society, Vol. 122, No. 2) 122 (2): 591–595, doi:10.2307/2161053, JSTOR 2161053

Chapter 32

Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, alongwith a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods. Thedefinition of a topological space relies only upon set theory and is the most general notion of a mathematical spacethat allows for the definition of concepts such as continuity, connectedness, and convergence.[1] Other spaces, such asmanifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being sogeneral, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics.The branch of mathematics that studies topological spaces in their own right is called point-set topology or generaltopology.

32.1 Definition

Main article: Characterizations of the category of topological spaces

The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure.Thus one chooses the axiomatisation suited for the application. The most commonly used, and the most elegant, isthat in terms of open sets, but the most intuitive is that in terms of neighbourhoods and so we give this first. Note: Avariety of more axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy.

32.1.1 Neighbourhoods definition

This axiomatization is due to Felix Hausdorff. Let X be a set; the elements of X are usually called points, though theycan be any mathematical object. We allow X to be empty. Let N be a function assigning to each x (point) in X anon-empty collection N(x) of subsets of X. The elements of N(x) will be called neighbourhoods of x with respect toN (or, simply, neighbourhoods of x). The function N is called a neighbourhood topology if the axioms below[2] aresatisfied; and then X with N is called a topological space.

1. If N is a neighbourhood of x (i.e., N ∈ N(x)), then x ∈ N. In other words, each point belongs to every one of itsneighbourhoods.

2. If N is a subset of X and contains a neighbourhood of x, then N is a neighbourhood of x. I.e., every supersetof a neighbourhood of a point x in X is again a neighbourhood of x.

3. The intersection of two neighbourhoods of x is a neighbourhood of x.

4. Any neighbourhood N of x contains a neighbourhood M of x such that N is a neighbourhood of each point ofM.

The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in thestructure of the theory, that of linking together the neighbourhoods of different points of X.

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80 CHAPTER 32. TOPOLOGICAL SPACE

A standard example of such a system of neighbourhoods is for the real line R, where a subset N of R is defined to bea neighbourhood of a real number x if there is an open interval containing x and contained in N.Given such a structure, we can define a subset U of X to be open if U is a neighbourhood of all points in U. It is aremarkable fact that the open sets then satisfy the elegant axioms given below, and that, given these axioms, we canrecover the neighbourhoods satisfying the above axioms by defining N to be a neighbourhood of x if N contains anopen set U such that x ∈ U.[3]

32.1.2 Open sets definition

1 2 3 1 2 3

1 2 3 1 2 3

1 2 3 1 2 3

Four examples and two non-examples of topologies on the three-point set 1,2,3. The bottom-left example is not a topology becausethe union of 2 and 3 [i.e. 2,3] is missing; the bottom-right example is not a topology because the intersection of 1,2 and2,3 [i.e. 2], is missing.

A topological space is then a set X together with a collection of subsets of X, called open sets and satisfying thefollowing axioms:[4]

1. The empty set and X itself are open.

2. Any union of open sets is open.

3. The intersection of any finite number of open sets is open.

The collection τ of open sets is then also called a topology onX, or, if more precision is needed, an open set topology.The sets in τ are called the open sets, and their complements in X are called closed sets. A subset of Xmay be neitherclosed nor open, either closed or open, or both. A set that is both closed and open is called a clopen set.

32.2. COMPARISON OF TOPOLOGIES 81

Examples

1. X = 1, 2, 3, 4 and collection τ = , 1, 2, 3, 4 of only the two subsets of X required by the axioms forma topology, the trivial topology (indiscrete topology).

2. X = 1, 2, 3, 4 and collection τ = , 2, 1, 2, 2, 3, 1, 2, 3, 1, 2, 3, 4 of six subsets of X formanother topology.

3. X = 1, 2, 3, 4 and collection τ = P(X) (the power set of X) form a third topology, the discrete topology.

4. X =Z, the set of integers, and collection τ equal to all finite subsets of the integers plusZ itself is not a topology,because (for example) the union of all finite sets not containing zero is infinite but is not all of Z, and so is notin τ .

32.1.3 Closed sets definition

Using de Morgan’s laws, the above axioms defining open sets become axioms defining closed sets:

1. The empty set and X are closed.

2. The intersection of any collection of closed sets is also closed.

3. The union of any pair of closed sets is also closed.

Using these axioms, another way to define a topological space is as a set X together with a collection τ of closedsubsets of X. Thus the sets in the topology τ are the closed sets, and their complements in X are the open sets.

32.1.4 Other definitions

There are many other equivalent ways to define a topological space: in other words, the concepts of neighbourhoodor of open respectively closed set can be reconstructed from other starting points and satisfy the correct axioms.Another way to define a topological space is by using the Kuratowski closure axioms, which define the closed sets asthe fixed points of an operator on the power set of X.A net is a generalisation of the concept of sequence. A topology is completely determined if for every net in X theset of its accumulation points is specified.

32.2 Comparison of topologies

Main article: Comparison of topologies

A variety of topologies can be placed on a set to form a topological space. When every set in a topology τ1 is also ina topology τ2 and τ1 is a subset of τ2, we say that τ2 is finer than τ1, and τ1 is coarser than τ2. A proof that reliesonly on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies onlyon certain sets not being open applies to any coarser topology. The terms larger and smaller are sometimes used inplace of finer and coarser, respectively. The terms stronger and weaker are also used in the literature, but with littleagreement on the meaning, so one should always be sure of an author’s convention when reading.The collection of all topologies on a given fixed set X forms a complete lattice: if F = τα| α in A is a collectionof topologies on X, then the meet of F is the intersection of F, and the join of F is the meet of the collection of alltopologies on X that contain every member of F.

32.3 Continuous functions

A function f : X→ Y between topological spaces is called continuous if for all x ∈X and all neighbourhoodsN of f(x)there is a neighbourhoodM of x such that f(M) ⊆N. This relates easily to the usual definition in analysis. Equivalently,

82 CHAPTER 32. TOPOLOGICAL SPACE

f is continuous if the inverse image of every open set is open.[5] This is an attempt to capture the intuition that thereare no “jumps” or “separations” in the function. A homeomorphism is a bijection that is continuous and whose inverseis also continuous. Two spaces are called homeomorphic if there exists a homeomorphism between them. From thestandpoint of topology, homeomorphic spaces are essentially identical.In category theory, Top, the category of topological spaces with topological spaces as objects and continuous functionsas morphisms is one of the fundamental categories in category theory. The attempt to classify the objects of thiscategory (up to homeomorphism) by invariants has motivated areas of research, such as homotopy theory, homologytheory, and K-theory etc.

32.4 Examples of topological spaces

A given set may have many different topologies. If a set is given a different topology, it is viewed as a differenttopological space. Any set can be given the discrete topology in which every subset is open. The only convergentsequences or nets in this topology are those that are eventually constant. Also, any set can be given the trivial topology(also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence andnet in this topology converges to every point of the space. This example shows that in general topological spaces,limits of sequences need not be unique. However, often topological spaces must be Hausdorff spaces where limitpoints are unique.There are many ways of defining a topology on R, the set of real numbers. The standard topology on R is generatedby the open intervals. The set of all open intervals forms a base or basis for the topology, meaning that every openset is a union of some collection of sets from the base. In particular, this means that a set is open if there exists anopen interval of non zero radius about every point in the set. More generally, the Euclidean spaces Rn can be givena topology. In the usual topology on Rn the basic open sets are the open balls. Similarly, C, the set of complexnumbers, and Cn have a standard topology in which the basic open sets are open balls.Every metric space can be given a metric topology, in which the basic open sets are open balls defined by the metric.This is the standard topology on any normed vector space. On a finite-dimensional vector space this topology is thesame for all norms.Many sets of linear operators in functional analysis are endowed with topologies that are defined by specifying whena particular sequence of functions converges to the zero function.Any local field has a topology native to it, and this can be extended to vector spaces over that field.Every manifold has a natural topology since it is locally Euclidean. Similarly, every simplex and every simplicialcomplex inherits a natural topology from Rn.The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic variety. On Rn or Cn, theclosed sets of the Zariski topology are the solution sets of systems of polynomial equations.A linear graph has a natural topology that generalises many of the geometric aspects of graphs with vertices andedges.The Sierpiński space is the simplest non-discrete topological space. It has important relations to the theory of com-putation and semantics.There exist numerous topologies on any given finite set. Such spaces are called finite topological spaces. Finite spacesare sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complementis finite. This is the smallest T1 topology on any infinite set.Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complementis countable. When the set is uncountable, this topology serves as a counterexample in many situations.The real line can also be given the lower limit topology. Here, the basic open sets are the half open intervals [a, b).This topology on R is strictly finer than the Euclidean topology defined above; a sequence converges to a point in thistopology if and only if it converges from above in the Euclidean topology. This example shows that a set may havemany distinct topologies defined on it.If Γ is an ordinal number, then the set Γ = [0, Γ) may be endowed with the order topology generated by the intervals(a, b), [0, b) and (a, Γ) where a and b are elements of Γ.

32.5. TOPOLOGICAL CONSTRUCTIONS 83

32.5 Topological constructions

Every subset of a topological space can be given the subspace topology in which the open sets are the intersectionsof the open sets of the larger space with the subset. For any indexed family of topological spaces, the product can begiven the product topology, which is generated by the inverse images of open sets of the factors under the projectionmappings. For example, in finite products, a basis for the product topology consists of all products of open sets. Forinfinite products, there is the additional requirement that in a basic open set, all but finitely many of its projectionsare the entire space.A quotient space is defined as follows: if X is a topological space and Y is a set, and if f : X→ Y is a surjectivefunction, then the quotient topology on Y is the collection of subsets of Y that have open inverse images under f. Inother words, the quotient topology is the finest topology on Y for which f is continuous. A common example of aquotient topology is when an equivalence relation is defined on the topological space X. The map f is then the naturalprojection onto the set of equivalence classes.The Vietoris topology on the set of all non-empty subsets of a topological space X, named for Leopold Vietoris, isgenerated by the following basis: for every n-tuple U1, ..., Un of open sets in X, we construct a basis set consistingof all subsets of the union of the Ui that have non-empty intersections with each Ui.

32.6 Classification of topological spaces

Topological spaces can be broadly classified, up to homeomorphism, by their topological properties. A topologicalproperty is a property of spaces that is invariant under homeomorphisms. To prove that two spaces are not home-omorphic it is sufficient to find a topological property not shared by them. Examples of such properties includeconnectedness, compactness, and various separation axioms.See the article on topological properties for more details and examples.

32.7 Topological spaces with algebraic structure

For any algebraic objects we can introduce the discrete topology, under which the algebraic operations are continuousfunctions. For any such structure that is not finite, we often have a natural topology compatible with the algebraicoperations, in the sense that the algebraic operations are still continuous. This leads to concepts such as topologicalgroups, topological vector spaces, topological rings and local fields.

32.8 Topological spaces with order structure

• Spectral. A space is spectral if and only if it is the prime spectrum of a ring (Hochster theorem).

• Specialization preorder. In a space the specialization (or canonical) preorder is defined by x ≤ y if andonly if clx ⊆ cly.

32.9 Specializations and generalizations

The following spaces and algebras are either more specialized or more general than the topological spaces discussedabove.

• Proximity spaces provide a notion of closeness of two sets.

• Metric spaces embody a metric, a precise notion of distance between points.

• Uniform spaces axiomatize ordering the distance between distinct points.

• A topological space in which the points are functions is called a function space.

84 CHAPTER 32. TOPOLOGICAL SPACE

• Cauchy spaces axiomatize the ability to test whether a net is Cauchy. Cauchy spaces provide a general settingfor studying completions.

• Convergence spaces capture some of the features of convergence of filters.

• Grothendieck sites are categories with additional data axiomatizing whether a family of arrows covers an object.Sites are a general setting for defining sheaves.

32.10 See also

• Space (mathematics)

• Kolmogorov space (T0)

• accessible/Fréchet space (T1)

• Hausdorff space (T2)

• Completely Hausdorff space and Urysohn space (T₂½)

• Regular space and regular Hausdorff space (T3)

• Tychonoff space and completely regular space (T₃½)

• Normal Hausdorff space (T4)

• Completely normal Hausdorff space (T5)

• Perfectly normal Hausdorff space (T6)

• Quasitopological space

• Complete Heyting algebra – The system of all open sets of a given topological space ordered by inclusion is acomplete Heyting algebra.

32.11 Notes[1] Schubert 1968, p. 13

[2] Brown 2006, section 2.1.

[3] Brown 2006, section 2.2.

[4] Armstrong 1983, definition 2.1.

[5] Armstrong 1983, theorem 2.6.

32.12 References

• Armstrong, M. A. (1983) [1979]. Basic Topology. Undergraduate texts in mathematics. Springer. ISBN0-387-90839-0.

• Bredon, Glen E., Topology and Geometry (Graduate Texts in Mathematics), Springer; 1st edition (October 17,1997). ISBN 0-387-97926-3.

• Bourbaki, Nicolas; Elements of Mathematics: General Topology, Addison-Wesley (1966).

• Brown, Ronald, Topology and groupoids, Booksurge (2006) ISBN 1-4196-2722-8 (3rd edition of differentlytitled books) (order from amazon.com).

• Čech, Eduard; Point Sets, Academic Press (1969).

32.13. EXTERNAL LINKS 85

• Fulton, William, Algebraic Topology, (Graduate Texts in Mathematics), Springer; 1st edition (September 5,1997). ISBN 0-387-94327-7.

• Lipschutz, Seymour; Schaum’s Outline of General Topology, McGraw-Hill; 1st edition (June 1, 1968). ISBN0-07-037988-2.

• Munkres, James; Topology, Prentice Hall; 2nd edition (December 28, 1999). ISBN 0-13-181629-2.

• Runde, Volker; A Taste of Topology (Universitext), Springer; 1st edition (July 6, 2005). ISBN 0-387-25790-X.

• Schubert, Horst (1968), Topology, Allyn and Bacon

• Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970).ISBN 0-03-079485-4.

• Vaidyanathaswamy, R. (1960). Set Topology. Chelsea Publishing Co. ISBN 0486404560.

• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

32.13 External links• Hazewinkel, Michiel, ed. (2001), “Topological space”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Topological space at PlanetMath.org.

Chapter 33

Topology

Not to be confused with topography.This article is about the branch of mathematics. For other uses, see Topology (disambiguation).In mathematics, topology (from the Greek τόπος, “place”, and λόγος, “study”), the study of topological spaces, is

Möbius strips, which have only one surface and one edge, are a kind of object studied in topology.

an area of mathematics concerned with the properties of space that are preserved under continuous deformations,such as stretching and bending, but not tearing or gluing. Important topological properties include connectedness andcompactness.Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space,dimension, and transformation. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned thegeometria situs (Greek-Latin for “geometry of place”) and analysis situs (Greek-Latin for “picking apart of place”).The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the firstdecades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century,topology had become a major branch of mathematics.Topology has many subfields:

• General topology establishes the foundational aspects of topology and investigates properties of topological

86

33.1. HISTORY 87

spaces and investigates concepts inherent to topological spaces. It includes point-set topology, which is thefoundational topology used in all other branches (including topics like compactness and connectedness).

• Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology andhomotopy groups.

• Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closelyrelated to differential geometry and together they make up the geometric theory of differentiable manifolds.

• Geometric topology primarily studies manifolds and their embeddings (placements) in other manifolds. Aparticularly active area is low dimensional topology, which studies manifolds of four or fewer dimensions.This includes knot theory, the study of mathematical knots.

A three-dimensional depiction of a thickened trefoil knot, the simplest non-trivial knot

See also: topology glossary for definitions of some of the terms used in topology, and topological space for a moretechnical treatment of the subject.

33.1 History

Topology began with the investigation of certain questions in geometry. Leonhard Euler's 1736 paper on the SevenBridges of Königsberg[1] is regarded as one of the first academic treatises in modern topology.The term “Topologie” was introduced in German in 1847 by Johann Benedict Listing in Vorstudien zur Topologie,[2]who had used the word for ten years in correspondence before its first appearance in print. The English form topologywas first used in 1883 in Listing’s obituary in the journal Nature[3] to distinguish “qualitative geometry from theordinary geometry in which quantitative relations chiefly are treated”. The term topologist in the sense of a specialistin topology was used in 1905 in the magazine Spectator. However, none of these uses corresponds exactly to themodern definition of topology.

88 CHAPTER 33. TOPOLOGY

The Seven Bridges of Königsberg was a problem solved by Euler.

Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19thcentury. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space aspart of his study of Fourier series.Henri Poincaré published Analysis Situs in 1895,[4] introducing the concepts of homotopy and homology, which arenow considered part of algebraic topology.Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, GiulioAscoli and others, Maurice Fréchet introduced the metric space in 1906.[5] A metric space is now considered aspecial case of a general topological space. In 1914, Felix Hausdorff coined the term “topological space” and gavethe definition for what is now called a Hausdorff space.[6] Currently, a topological space is a slight generalization ofHausdorff spaces, given in 1922 by Kazimierz Kuratowski.For further developments, see point-set topology and algebraic topology.

33.2 Introduction

Topology can be formally defined as “the study of qualitative properties of certain objects (called topological spaces)that are invariant under a certain kind of transformation (called a continuous map), especially those properties thatare invariant under a certain kind of transformation (called homeomorphism).”Topology is also used to refer to a structure imposed upon a set X, a structure that essentially 'characterizes’ the set Xas a topological space by taking proper care of properties such as convergence, connectedness and continuity, upontransformation.Topological spaces show up naturally in almost every branch of mathematics. This has made topology one of thegreat unifying ideas of mathematics.

33.2. INTRODUCTION 89

The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objectsinvolved, but rather on the way they are put together. For example, the square and the circle have many properties incommon: they are both one dimensional objects (from a topological point of view) and both separate the plane intotwo parts, the part inside and the part outside.One of the first papers in topology was the demonstration, by Leonhard Euler, that it was impossible to find a routethrough the town of Königsberg (now Kaliningrad) that would cross each of its seven bridges exactly once. Thisresult did not depend on the lengths of the bridges, nor on their distance from one another, but only on connectivityproperties: which bridges are connected to which islands or riverbanks. This problem in introductory mathematicscalled Seven Bridges of Königsberg led to the branch of mathematics known as graph theory.

A continuous deformation (a type of homeomorphism) of a mug into a doughnut (torus) and back

Similarly, the hairy ball theorem of algebraic topology says that “one cannot comb the hair flat on a hairy ball withoutcreating a cowlick.” This fact is immediately convincing to most people, even though they might not recognize themore formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere.As with the Bridges of Königsberg, the result does not depend on the shape of the sphere; it applies to any kind ofsmooth blob, as long as it has no holes.To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just whatproperties these problems do rely on. From this need arises the notion of homeomorphism. The impossibility ofcrossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and thehairy ball theorem applies to any space homeomorphic to a sphere.

90 CHAPTER 33. TOPOLOGY

Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditionaljoke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut couldbe reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.Homeomorphism can be considered the most basic topological equivalence. Another is homotopy equivalence. Thisis harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent ifthey both result from “squishing” some larger object.An introductory exercise is to classify the uppercase letters of the English alphabet according to homeomorphismand homotopy equivalence. The result depends partially on the font used. The figures use the sans-serif Myriad font.Homotopy equivalence is a rougher relationship than homeomorphism; a homotopy equivalence class can containseveral homeomorphism classes. The simple case of homotopy equivalence described above can be used here toshow two letters are homotopy equivalent. For example, O fits inside P and the tail of the P can be squished to the“hole” part.Homeomorphism classes are:

• no holes,

• no holes three tails,

• no holes four tails,

• one hole no tail,

• one hole one tail,

• one hole two tails,

• two holes no tail, and

• a bar with four tails (the “bar” on the K is almost too short to see).

Homotopy classes are larger, because the tails can be squished down to a point. They are:

• one hole,

• two holes, and

• no holes.

To be sure that the letters are classified correctly, we need to show that two letters in the same class are equivalent andtwo letters in different classes are not equivalent. In the case of homeomorphism, this can be done by selecting pointsand showing their removal disconnects the letters differently. For example, X and Y are not homeomorphic becauseremoving the center point of the X leaves four pieces; whatever point in Y corresponds to this point, its removalcan leave at most three pieces. The case of homotopy equivalence is harder and requires a more elaborate argumentshowing an algebraic invariant, such as the fundamental group, is different on the supposedly differing classes.Letter topology has practical relevance in stencil typography. For instance, Braggadocio font stencils are made of oneconnected piece of material.

33.3 Concepts

33.3.1 Topologies on Sets

Main article: Topological space

The term topology also refers to a specific mathematical idea which is central to the area of mathematics calledtopology. Informally, a topology is used to tell how elements of a set are related spatially to each other. The same setcan have different topologies. For instance, the real line, the complex plane, and the Cantor set can be thought of asthe same set with different topologies.Formally, let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:

33.4. TOPICS 91

1. Both the empty set and X are elements of τ

2. Any union of elements of τ is an element of τ

3. Any intersection of finitely many elements of τ is an element of τ

If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation Xτmay be used to denote a setX endowed with the particular topology τ.The members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ (i.e., itscomplement is open). A subset of X may be open, closed, both (clopen set), or neither. The empty set and X itselfare always both closed and open. An open set containing a point x is called a 'neighborhood' of x.A set with a topology is called a topological space.

33.3.2 Continuous functions and homeomorphisms

Main articles: Continuous function and homeomorphism

A function or map from one topological space to another is called continuous if the inverse image of any open set isopen. If the function maps the real numbers to the real numbers (both spaces with the Standard Topology), then thisdefinition of continuous is equivalent to the definition of continuous in calculus. If a continuous function is one-to-oneand onto, and if the inverse of the function is also continuous, then the function is called a homeomorphism and thedomain of the function is said to be homeomorphic to the range. Another way of saying this is that the function hasa natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties, andare considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and thedoughnut. But the circle is not homeomorphic to the doughnut.

33.3.3 Manifolds

Main article: Manifold

While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiarclass of spaces known as manifolds. A manifold is a topological space that resembles Euclidean space near eachpoint. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to theEuclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can allbe realized in three dimensions, but also the Klein bottle and real projective plane which cannot.

33.4 Topics

33.4.1 General topology

Main article: General topology

General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions usedin topology.[7][8] It is the foundation of most other branches of topology, including differential topology, geometrictopology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness. Intuitively, continu-ous functions take nearby points to nearby points; compact sets are those which can be covered by finitely many setsof arbitrarily small size; and connected sets are sets which cannot be divided into two pieces which are far apart. Thewords 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. Ifwe change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are.Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.

92 CHAPTER 33. TOPOLOGY

Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric.Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.

33.4.2 Algebraic topology

Main article: Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.[9]The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usuallymost classify up to homotopy equivalence.The most important of these invariants are homotopy groups, homology, and cohomology.Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraicproblems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroupof a free group is again a free group.

33.4.3 Differential topology

Main article: Differential topology

Differential topology is the field dealing with differentiable functions on differentiable manifolds.[10] It is closelyrelated to differential geometry and together they make up the geometric theory of differentiable manifolds.More specifically, differential topology considers the properties and structures that require only a smooth structureon a manifold to be defined. Smooth manifolds are 'softer' than manifolds with extra geometric structures, whichcan act as obstructions to certain types of equivalences and deformations that exist in differential topology. Forinstance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on thesame smooth manifold—that is, one can smoothly “flatten out” certain manifolds, but it might require distorting thespace and affecting the curvature or volume.

33.4.4 Geometric topology

Main article: Geometric topology

Geometric topology is a branch of topology that primarily focuses on low-dimensional manifolds (i.e. dimensions2,3 and 4) and their interaction with geometry, but it also includes some higher-dimensional topology.[11] [12] Someexamples of topics in geometric topology are orientability, handle decompositions, local flatness, and the planar andhigher-dimensional Schönflies theorem.In high-dimensional topology, characteristic classes are a basic invariant, and surgery theory is a key theory.Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – ev-ery surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curva-ture/spherical, zero curvature/flat, negative curvature/hyperbolic – and the geometrization conjecture (now theorem)in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of 8 possible geometries.2-dimensional topology can be studied as complex geometry in one variable (Riemann surfaces are complex curves)– by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces),though not every 4-manifold admits a complex structure.

33.4.5 Generalizations

Occasionally, one needs to use the tools of topology but a “set of points” is not available. In pointless topology oneconsiders instead the lattice of open sets as the basic notion of the theory,[13] while Grothendieck topologies are

33.5. APPLICATIONS 93

structures defined on arbitrary categories that allow the definition of sheaves on those categories, and with that thedefinition of general cohomology theories.[14]

33.5 Applications

33.5.1 Biology

Knot theory, a branch of topology, is used in biology to study the effects of certain enzymes on DNA. These enzymescut, twist, and reconnect the DNA, causing knotting with observable effects such as slower electrophoresis.[15] Topol-ogy is also used in evolutionary biology to represent the relationship between phenotype and genotype.[16] Phenotypicforms which appear quite different can be separated by only a few mutations depending on how genetic changes mapto phenotypic changes during development.

33.5.2 Computer science

Topological data analysis uses techniques from algebraic topology to determine the large scale structure of a set (forinstance, determining if a cloud of points is spherical or toroidal). The main method used by topological data analysisis:

1. Replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter.

2. Analyse these topological complexes via algebraic topology — specifically, via the theory of persistent homol-ogy.[17]

3. Encode the persistent homology of a data set in the form of a parameterized version of a Betti number whichis called a barcode.[17]

33.5.3 Physics

In physics, topology is used in several areas such as quantum field theory and cosmology.A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which com-putes topological invariants.Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among otherthings, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces inalgebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topo-logical field theory.In cosmology, topology can be used to describe the overall shape of the universe.[18] This area is known as spacetimetopology.

33.5.4 Robotics

The various possible positions of a robot can be described by a manifold called configuration space.[19] In the area ofmotion planning, one finds paths between two points in configuration space. These paths represent a motion of therobot’s joints and other parts into the desired location and pose.

33.6 See also

• Equivariant topology

• General topology

• List of algebraic topology topics

94 CHAPTER 33. TOPOLOGY

• List of examples in general topology

• List of general topology topics

• List of geometric topology topics

• List of topology topics

• Publications in topology

• Topology glossary

33.7 References

[1] Euler, Leonhard, Solutio problematis ad geometriam situs pertinentis

[2] Listing, Johann Benedict, “Vorstudien zur Topologie”, Vandenhoeck und Ruprecht, Göttingen, p. 67, 1848

[3] Tait, Peter Guthrie, “Johann Benedict Listing (obituary)", Nature *27*, 1 February 1883, pp. 316–317

[4] Poincaré, Henri, “Analysis situs”, Journal de l'École Polytechnique ser 2, 1 (1895) pp. 1–123

[5] Fréchet, Maurice, “Sur quelques points du calcul fonctionnel”, PhD dissertation, 1906

[6] Hausdorff, Felix, “Grundzüge der Mengenlehre”, Leipzig: Veit. In (Hausdorff Werke, II (2002), 91–576)

[7] Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.

[8] Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall,2008.

[9] Allen Hatcher, Algebraic topology. (2002) Cambridge University Press, xii+544 pp. ISBN 0-521-79160-X and ISBN0-521-79540-0.

[10] Lee, John M. (2006). Introduction to Smooth Manifolds. Springer-Verlag. ISBN 978-0-387-95448-6.

[11] Budney, Ryan (2011). “What is geometric topology?". mathoverflow.net. Retrieved 29 December 2013.

[12] R.B. Sher and R.J. Daverman (2002), Handbook of Geometric Topology, North-Holland. ISBN 0-444-82432-4

[13] Johnstone, Peter T., 1983, "The point of pointless topology," Bulletin of the American Mathematical Society 8(1): 41-53.

[14] Artin, Michael (1962). Grothendieck topologies. Cambridge, MA: Harvard University, Dept. of Mathematics. Zbl0208.48701.

[15] Adams, Colin (2004). The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Math-ematical Society. ISBN 0-8218-3678-1

[16] Barble M R Stadler et al. “The Topology of the Possible: Formal Spaces Underlying Patterns of Evolutionary Change”.Journal of Theoretical Biology 213: 241–274. doi:10.1006/jtbi.2001.2423.

[17] Gunnar Carlsson (April 2009). “Topology and data” (PDF). BULLETIN (New Series) OF THE AMERICAN MATHEMAT-ICAL SOCIETY 46 (2): 255–308. doi:10.1090/S0273-0979-09-01249-X.

[18] The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds 2nd ed (Marcel Dekker, 1985, ISBN0-8247-7437-X)

[19] John J. Craig, Introduction to Robotics: Mechanics and Control, 3rd Ed. Prentice-Hall, 2004

33.8. FURTHER READING 95

33.8 Further reading• Ryszard Engelking, General Topology, Heldermann Verlag, Sigma Series in Pure Mathematics, December1989, ISBN 3-88538-006-4.

• Bourbaki; Elements of Mathematics: General Topology, Addison–Wesley (1966).

• Breitenberger, E. (2006). “Johann Benedict Listing”. In James, I. M. History of Topology. North Holland.ISBN 978-0-444-82375-5.

• Kelley, John L. (1975). General Topology. Springer-Verlag. ISBN 0-387-90125-6.

• Brown, Ronald (2006). Topology and Groupoids. Booksurge. ISBN 1-4196-2722-8. (Provides a well mo-tivated, geometric account of general topology, and shows the use of groupoids in discussing van Kampen’stheorem, covering spaces, and orbit spaces.)

• Wacław Sierpiński, General Topology, Dover Publications, 2000, ISBN 0-486-41148-6

• Pickover, Clifford A. (2006). The Möbius Strip: Dr. August Möbius’s Marvelous Band in Mathematics, Games,Literature, Art, Technology, and Cosmology. Thunder’s Mouth Press. ISBN 1-56025-826-8. (Provides apopular introduction to topology and geometry)

• Gemignani, Michael C. (1990) [1967], Elementary Topology (2nd ed.), Dover Publications Inc., ISBN 0-486-66522-4

33.9 External links• Hazewinkel, Michiel, ed. (2001), “Topology, general”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Elementary Topology: A First Course Viro, Ivanov, Netsvetaev, Kharlamov.

• Topology at DMOZ

• The Topological Zoo at The Geometry Center.

• Topology Atlas

• Topology Course Lecture Notes Aisling McCluskey and Brian McMaster, Topology Atlas.

• Topology Glossary

• Moscow 1935: Topology moving towards America, a historical essay by Hassler Whitney.

Chapter 34

Totally bounded space

In topology and related branches of mathematics, a totally bounded space is a space that can be covered by finitelymany subsets of any fixed “size” (where the meaning of “size” depends on the given context). The smaller the sizefixed, the more subsets may be needed, but any specific size should require only finitely many subsets. A relatednotion is a totally bounded set, in which only a subset of the space needs to be covered. Every subset of a totallybounded space is a totally bounded set; but even if a space is not totally bounded, some of its subsets still will be.The term precompact (or pre-compact) is sometimes used with the same meaning, but `pre-compact' is also usedto mean relatively compact. In a complete metric space these meanings coincide but in general they do not. See alsouse of the axiom of choice below.

34.1 Definition for a metric space

A metric space (M,d) is totally bounded if and only if for every real number ϵ > 0 , there exists a finite collectionof open balls inM of radius ϵ whose union containsM . Equivalently, the metric spaceM is totally bounded if andonly if for every ϵ > 0 , there exists a finite cover such that the radius of each element of the cover is at most ϵ . Thisis equivalent to the existence of a finite ε-net.[1]

Each totally bounded space is bounded (as the union of finitely many bounded sets is bounded), but the converse isnot true in general. For example, an infinite set equipped with the discrete metric is bounded but not totally bounded.If M is Euclidean space and d is the Euclidean distance, then a subset (with the subspace topology) is totally boundedif and only if it is bounded.A metric space is said to be precompact if every sequence admits a Cauchy subsequence. Thus for metric spaceswe have: compactness = precompactness + completeness. It turns out that the space is precompact if and only if it istotally bounded. Therefore both names can be used interchangeably.

34.2 Definitions in other contexts

The general logical form of the definition is: a subset S of a space X is a totally bounded set if and only if, given anysize E, there exist a natural number n and a family A1, A2, ..., An of subsets of X, such that S is contained in the unionof the family (in other words, the family is a finite cover of S), and such that each set Ai in the family is of size E (orless). In mathematical symbols:

∀E ∃n ∈ N , A1, A2, . . . , An ⊆ X

(S ⊆

n∪i=1

Ai and ∀i = 1, . . . , n size(Ai) ≤ E

).

The space X is a totally bounded space if and only if it is a totally bounded set when considered as a subset of itself.(One can also define totally bounded spaces directly, and then define a set to be totally bounded if and only if it istotally bounded when considered as a subspace.)

96

34.3. EXAMPLES AND NONEXAMPLES 97

The terms “space” and “size” here are vague, and they may be made precise in various ways:A subset S of a metric space X is totally bounded if and only if, given any positive real number E, there exists a finitecover of S by subsets of X whose diameters are all less than E. (In other words, a “size” here is a positive real number,and a subset is of size E if its diameter is less than E.) Equivalently, S is totally bounded if and only if, given any Eas before, there exist elements a1, a2, ..., an of X such that S is contained in the union of the n open balls of radius Earound the points ai.A subset S of a topological vector space, or more generally topological abelian group, X is totally bounded if and onlyif, given any neighbourhood E of the identity (zero) element of X, there exists a finite cover of S by subsets of X eachof which is a translate of a subset of E. (In other words, a “size” here is a neighbourhood of the identity element, anda subset is of size E if it is translate of a subset of E.) Equivalently, S is totally bounded if and only if, given any E asbefore, there exist elements a1, a2, ..., an of X such that S is contained in the union of the n translates of E by thepoints ai.A topological group X is left-totally bounded if and only if it satisfies the definition for topological abelian groupsabove, using left translates. That is, use aiE in place of E + ai. Alternatively, X is right-totally bounded if and only ifit satisfies the definition for topological abelian groups above, using right translates. That is, use Eai in place of E +ai. (In other words, a “size” here is unambiguously a neighbourhood of the identity element, but there are two notionsof whether a set is of a given size: a left notion based on left translation and a right notion based on right translation.)Generalising the above definitions, a subset S of a uniform space X is totally bounded if and only if, given anyentourage E in X, there exists a finite cover of S by subsets of X each of whose Cartesian squares is a subset ofE. (In other words, a “size” here is an entourage, and a subset is of size E if its Cartesian square is a subset of E.)Equivalently, S is totally bounded if and only if, given any E as before, there exist subsets A1, A2, ..., An of X suchthat S is contained in the union of the Ai and, whenever the elements x and y of X both belong to the same set Ai,then (x,y) belongs to E (so that x and y are close as measured by E).The definition can be extended still further, to any category of spaces with a notion of compactness and Cauchycompletion: a space is totally bounded if and only if its completion is compact.

34.3 Examples and nonexamples• A subset of the real line, or more generally of (finite-dimensional) Euclidean space, is totally bounded if andonly if it is bounded. Archimedean property is used.

• The unit ball in a Hilbert space, or more generally in a Banach space, is totally bounded if and only if the spacehas finite dimension.

• Every compact set is totally bounded, whenever the concept is defined.

• Every totally bounded metric space is bounded. However not every bounded metric space is totally bounded.[2]

• A subset of a complete metric space is totally bounded if and only if it is relatively compact (meaning that itsclosure is compact).

• In a locally convex space endowed with the weak topology the precompact sets are exactly the bounded sets.

• A metric space is separable if and only if it is homeomorphic to a totally bounded metric space.[2]

• An infinite metric space with the discrete metric (the distance between any two distinct points is 1) is not totallybounded, even though it is bounded.

34.4 Relationships with compactness and completeness

There is a nice relationship between total boundedness and compactness:Every compact metric space is totally bounded.A uniform space is compact if and only if it is both totally bounded and Cauchy complete. This can be seen as ageneralisation of the Heine–Borel theorem from Euclidean spaces to arbitrary spaces: we must replace boundednesswith total boundedness (and also replace closedness with completeness).

98 CHAPTER 34. TOTALLY BOUNDED SPACE

There is a complementary relationship between total boundedness and the process of Cauchy completion: A uniformspace is totally bounded if and only if its Cauchy completion is totally bounded. (This corresponds to the fact that,in Euclidean spaces, a set is bounded if and only if its closure is bounded.)Combining these theorems, a uniform space is totally bounded if and only if its completion is compact. This maybe taken as an alternative definition of total boundedness. Alternatively, this may be taken as a definition of pre-compactness, while still using a separate definition of total boundedness. Then it becomes a theorem that a space istotally bounded if and only if it is precompact. (Separating the definitions in this way is useful in the absence of theaxiom of choice; see the next section.)

34.5 Use of the axiom of choice

The properties of total boundedness mentioned above rely in part on the axiom of choice. In the absence of theaxiom of choice, total boundedness and precompactness must be distinguished. That is, we define total boundednessin elementary terms but define precompactness in terms of compactness and Cauchy completion. It remains true (thatis, the proof does not require choice) that every precompact space is totally bounded; in other words, if the completionof a space is compact, then that space is totally bounded. But it is no longer true (that is, the proof requires choice)that every totally bounded space is precompact; in other words, the completion of a totally bounded space might notbe compact in the absence of choice.

34.6 See also• Measure of non-compactness

• Locally compact space

34.7 Notes[1] Sutherland p.139

[2] Willard, p. 182

34.8 References• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

• Sutherland, W.A. (1975). Introduction to metric and topological spaces. Oxford University Press. ISBN 0-19-853161-3. Zbl 0304.54002.

Chapter 35

σ-compact space

In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces.[1]

A space is said to be σ-locally compact if it is both σ-compact and locally compact.[2]

35.1 Properties and examples

• Every compact space is σ-compact, and every σ-compact space is Lindelöf (i.e. every open cover has a count-able subcover).[3] The reverse implications do not hold, for example, standard Euclidean space (Rn) is σ-compact but not compact,[4] and the lower limit topology on the real line is Lindelöf but not σ-compact.[5] Infact, the countable complement topology is Lindelöf but neither σ-compact nor locally compact.[6]

• A Hausdorff, Baire space that is also σ-compact, must be locally compact at at least one point.

• If G is a topological group and G is locally compact at one point, then G is locally compact everywhere.Therefore, the previous property tells us that if G is a σ-compact, Hausdorff topological group that is also aBaire space, then G is locally compact. This shows that for Hausdorff topological groups that are also Bairespaces, σ-compactness implies local compactness.

• The previous property implies for instance thatRω is not σ-compact: if it were σ-compact, it would necessarilybe locally compact since Rω is a topological group that is also a Baire space.

• Every hemicompact space is σ-compact.[7] The converse, however, is not true;[8] for example, the space ofrationals, with the usual topology, is σ-compact but not hemicompact.

• The product of a finite number of σ-compact spaces is σ-compact. However the product of an infinite numberof σ-compact spaces may fail to be σ-compact.[9]

• A σ-compact space X is second category (resp. Baire) if and only if the set of points at which is X is locallycompact is nonempty (resp. dense) in X.[10]

35.2 See also

• Exhaustion by compact sets

• Lindelöf space

99

100 CHAPTER 35. Σ-COMPACT SPACE

35.3 Notes[1] Steen, p.19; Willard, p. 126.

[2] Steen, p. 21.

[3] Steen, p. 19.

[4] Steen, p. 56.

[5] Steen, p. 75–76.

[6] Steen, p. 50.

[7] Willard, p. 126.

[8] Willard, p. 126.

[9] Willard, p. 126.

[10] Willard, p. 188.

35.4 References• Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970).ISBN 0-03-079485-4.

• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

35.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 101

35.5 Text and image sources, contributors, and licenses

35.5.1 Text• A-paracompact space Source: http://en.wikipedia.org/wiki/A-paracompact_space?oldid=532052977 Contributors: Zundark, Paul Au-

gust, Vipul, Michael Slone, SmackBot, Silly rabbit, Lambiam, Cydebot, Ryan, JackSchmidt, Addbot, Brad7777 and Anonymous: 3

• Closure (topology) Source: http://en.wikipedia.org/wiki/Closure_(topology)?oldid=661885869 Contributors: AxelBoldt, Zundark, TheAnome, Toby Bartels, Edemaine, Michael Hardy, Wshun, Andres, Revolver, Charles Matthews, Zoicon5, Grendelkhan, MathMartin,Tobias Bergemann, Tosha, Giftlite, Rich Farmbrough, TedPavlic, Paul August, BenjBot, El C, HasharBot~enwiki, Sligocki, Oleg Alexan-drov, Isnow, Grammarbot, Jshadias, Chobot, BOT-Superzerocool, Banus, Zvika, Selfworm, Melchoir, RuudVisser, Dreadstar, NeilFraser,Gregbard, Salgueiro~enwiki, JAnDbot, Sullivan.t.j, Daniele.tampieri, Policron, Egaida, TXiKiBoT, VVVBot, Reinderien, Veddharta,Outhwest, Thehotelambush, Kumioko, Anchor Link Bot, Addbot, Yobot, Ht686rg90, Calle, Erel Segal, Ciphers, ArthurBot, Xqbot,Erik9bot, DrilBot, EmausBot, ZéroBot, JordiGH, Wcherowi, YFdyh-bot, Tarpuq and Anonymous: 23

• Compact operator Source: http://en.wikipedia.org/wiki/Compact_operator?oldid=664484742 Contributors: Michael Hardy, Takuya-Murata, Charles Matthews, Prumpf, AndrewKepert, Topbanana, Giftlite, BenFrantzDale, Lethe, Lupin, Billlion, LutzL, Oleg Alexan-drov, Linas, Igny, Mathbot, YurikBot, Wavelength, Scineram, Zvika, SmackBot, Silly rabbit, Tsca.bot, Daqu, Tac-Tics, Mct mht, WISo,Eleuther, Salgueiro~enwiki, Sullivan.t.j, STBotD, Kyle the bot, M gol, Jmath666, Biscuittin, Zsuoyb, Addbot, Lightbot, TaBOT-zerem,AnomieBOT, Citation bot, Bdmy, Citation bot 1, John of Reading, Drusus 0, Bomazi, Minimalrho, Mgkrupa, SoSivr and Anonymous:16

• Compact space Source: http://en.wikipedia.org/wiki/Compact_space?oldid=663827307 Contributors: AxelBoldt, Zundark, Andre En-gels, Toby~enwiki, Toby Bartels, Youandme, Patrick, Michael Hardy, Dominus, Firebirth, TakuyaMurata, BenKovitz, Revolver, CharlesMatthews, Dcoetzee, Dysprosia, Jitse Niesen, Zoicon5, Rik Bos, Lumos3, Robbot, MathMartin, Aetheling, Fuelbottle, Tobias Bergemann,Weialawaga~enwiki, Tosha, Giftlite, Markus Krötzsch, Lethe, Fropuff, Dratman, Matt Crypto, Python eggs, DRMacIver, Vivacissama-mente, Pyrop, Rich Farmbrough, TedPavlic, Guanabot, Paul August, El C, Rgdboer, Andi5, Vipul, Teorth, Kjkolb, Dbastos~enwiki,JohnyDog, Eric Kvaalen, Caesura, Oleg Alexandrov, Gmaxwell, Linas, Blair P. Houghton, OdedSchramm, Neocapitalist, Dionyziz, Gra-ham87, Qwertyus, Саша Стефановић, GeoffO, Mike Segal, Mathbot, Sodin, Chobot, Algebraist, YurikBot, Eraserhead1, RussBot,Lenthe, Trovatore, Crasshopper, ManoaChild, Bota47, Ms2ger, Edin1, Eigenlambda, Sardanaphalus, SmackBot, BeteNoir, Incnis Mrsi,Slaniel, Silly rabbit, DHN-bot~enwiki, Tekhnofiend, RFightmaster, Daqu, SashatoBot, Gandalfxviv, Landonproctor, ALife~enwiki, FellCollar, JRSpriggs, Sniffnoy, Sabate, Cydebot, Headbomb, Dbeatty, Futurebird, JAnDbot, Skimnc, Sullivan.t.j, David Eppstein, R'n'B,J.delanoy, Numbo3, Maurice Carbonaro, TomyDuby, Trumpet marietta 45750, Funandtrvl, LokiClock, PMajer, Plclark, Wikimorphism,FMasic, YohanN7, SieBot, JackSchmidt, Deadlyhair, Fakhredinblog, Roed314, Mpd1989, Lartoven, Cenarium, Hans Adler, Lkruijsw,Mathematix, Humanengr, Marc van Leeuwen, SilvonenBot, Algebran, Topology Expert, Lightbot, Matěj Grabovský, Legobot, Luckas-bot, Ht686rg90, Kilom691, Compsonheir, Erel Segal, Ciphers, Citation bot, ArthurBot, Bdmy, Roquedias, Veltas, Point-set topologist,FrescoBot, Sławomir Biały, Citation bot 1, Tkuvho, Rausch, Tim1357, Bhanin, Trappist the monk, 777sms, Thomassteinke, Rjwilmsi-Bot, EmausBot, Dadaist6174, Fly by Night, GoingBatty, Slawekb, Bethnim, AvicBot, Chharvey, D.Lazard, Hanne v, Zstyron, Joel B.Lewis, MerlIwBot, Helpful Pixie Bot, Agemineye, Shivsagardharam, Langing, BG19bot, Paolo Lipparini, TricksterWolf, Perspectiva8,Mureebe, Kuthikuthikuthi, Brirush, Tducote, Mark viking, Mathmensch, Omertamuz and Anonymous: 123

• Compactly embedded Source: http://en.wikipedia.org/wiki/Compactly_embedded?oldid=607163034 Contributors: Toby Bartels, Ben-FrantzDale, Silly rabbit, JAnDbot, Sullivan.t.j, Jmath666, AlleborgoBot, Ulrigo, Dingenis, Addbot, Yobot, Citation bot, Sławomir Biały,Citation bot 1 and Anonymous: 1

• Cover (topology) Source: http://en.wikipedia.org/wiki/Cover_(topology)?oldid=608317742Contributors: Zundark,Matusz, Tobias Berge-mann, Nonick, Fropuff, David Schaich, Paul August, EmilJ, Jumbuck, ABCD, Salix alba, Chobot, Roboto de Ajvol, YurikBot, Jsnx,Yliumath, Mhss, Dreadstar, Julian Mendez, Histwr~enwiki, RebelRobot, Policron, VolkovBot, CWii, LokiClock, Synthebot, Enough-said05, Qwfp, Silvercromagnon, SilvonenBot, Addbot, Topology Expert, Ago68, Numbo3-bot, Luckas-bot, Ht686rg90, Ciphers, Xqbot,RibotBOT, FrescoBot, LucienBOT, Petitlait, EmausBot, WikitanvirBot, DeathOfBalance, Noix07 and Anonymous: 24

• Exhaustion by compact sets Source: http://en.wikipedia.org/wiki/Exhaustion_by_compact_sets?oldid=379469145 Contributors: Zun-dark, TakuyaMurata, Charles Matthews, Giftlite, Oleg Alexandrov, Catamorphism, SmackBot, Silly rabbit, R'n'B, MikeRumex andAnonymous: 3

• Feebly compact space Source: http://en.wikipedia.org/wiki/Feebly_compact_space?oldid=626310065 Contributors: Zundark, Vipul,Hennobrandsma, Silly rabbit, Vanish2, David Eppstein and Yobot

• Functional analysis Source: http://en.wikipedia.org/wiki/Functional_analysis?oldid=659336520Contributors: AxelBoldt, Zundark, Youandme,Stevertigo, Michael Hardy, Kku, BenKovitz, Rotem Dan, Revolver, Charles Matthews, Dysprosia, Jitse Niesen, Fuzheado, Phys, Bevo,Robbot, Humus sapiens, Tobias Bergemann, Pdenapo,Weialawaga~enwiki, Giftlite, Lethe, Lupin, Fropuff, Ssd, Prosfilaes, DefLog~enwiki,AmarChandra, Abar, Paul August, Bender235, Billlion, Brian0918, Msh210, Derbeth, Oleg Alexandrov, Joriki, Woohookitty, Linas, Igny,Ae-a, Ruud Koot, Grammarbot, Koavf, Mathbot, John Z, YurikBot, Froth, BOT-Superzerocool, Banus, Lunch, Finell, Sardanaphalus,SmackBot, Alsandro, MalafayaBot, Silly rabbit, DHN-bot~enwiki, Can't sleep, clown will eat me, Cwzwarich, Allan McInnes, Merge,Richard L. Peterson, Kevin Murray, CRGreathouse, Yaris678, KennyDC, Thenub314, Magioladitis, RogierBrussee, Althai, Awake-forever, Allstarecho, Jtir, 1000Faces, Policron, VolkovBot, Camrn86, Kyle the bot, Temurjin, TXiKiBoT, Don4of4, Jmath666, Alle-borgoBot, Quietbritishjim, SieBot, Stca74, Brews ohare, DumZiBoT, WikHead, D.M. from Ukraine, Numbo3-bot, Lightbot, Legobot,Luckas-bot, Time Dilation, ArthurBot, Bdmy, Supernova0, Charvest, CES1596, FrescoBot, Kiefer.Wolfowitz, RedBot, Tweet7, SepIHw,Jowa fan, EmausBot, JJasper123, Mathuvw, JFB80, Maxdlink, ClueBot NG, BG19bot, Brad7777, Danwizard208, Randomguess, Kodi-ologist, Brirush, SakeUPenn, Mgkrupa, SoSivr and Anonymous: 85

• H-closed space Source: http://en.wikipedia.org/wiki/H-closed_space?oldid=594778406 Contributors: Paolo Lipparini and Anonymous:2

• Hemicompact space Source: http://en.wikipedia.org/wiki/Hemicompact_space?oldid=631224234 Contributors: Charles Matthews, To-bias Bergemann, D6, Paul August, Vipul, CommandoGuard~enwiki, Silly rabbit, Cydebot, Vanish2, Addbot, LucienBOT, Mgkrupa andAnonymous: 3

102 CHAPTER 35. Σ-COMPACT SPACE

• Interior (topology) Source: http://en.wikipedia.org/wiki/Interior_(topology)?oldid=659062423 Contributors: Mav, The Anome, TobyBartels, TakuyaMurata, GTBacchus, Revolver, Charles Matthews, Dysprosia, Tosha, Giftlite, Jason Quinn, Yuval madar, EmilJ, Ole-galexandrov, Oleg Alexandrov, Linas, Isnow, Mike Segal, Juan Marquez, Margosbot~enwiki, RexNL, Kri, Chobot, Roboto de Ajvol,YurikBot, Splash, Gwaihir, Pred, Poulpy, Banus, Zvika, Selfworm, BiT, JAn Dudík, Bluebot, Dreadstar, Mwtoews, Lambiam, Digana,Madmath789, Vaughan Pratt, Iokseng, JAnDbot, Mathematrucker, SieBot, Thehotelambush, Anchor Link Bot, Carolus m, BOTarate,Addbot, CarsracBot, Luckas-bot, TaBOT-zerem, Erel Segal, Ciphers, MorphismOfDoom, Zfeinst, Chewings72, Stephan Kulla, Mgkrupaand Anonymous: 24

• K-cell (mathematics) Source: http://en.wikipedia.org/wiki/K-cell_(mathematics)?oldid=648451922 Contributors: Michael Hardy, JoeDecker, Gilliam, KathrynLybarger, Rankersbo, Alvin Seville, Paulschn, Brirush, Ncandido, Nkrish96 and Anonymous: 3

• Lebesgue covering dimension Source: http://en.wikipedia.org/wiki/Lebesgue_covering_dimension?oldid=660515923Contributors: Ax-elBoldt, Zundark, The Anome, Charles Matthews, MathMartin, Tobias Bergemann, Tosha, Jason Quinn, Eequor, Kuratowski’s Ghost,Cogent, DiegoMoya, OlegAlexandrov, Joriki, Linas, OdedSchramm,Mathbot, YurikBot, Trovatore, RL0919, SmackBot, Mohan1986, AGeek Tragedy, Kuru, Noegenesis, JMK, Mulder416sBot, CBM, Mct mht, VictorAnyakin, Hut 8.5, David Eppstein, Policron, Sigmundur,Izno, TXiKiBoT, PipepBot, Grubb257, Alexbot, 1ForTheMoney, DumZiBoT, Addbot, 5 albert square, Luckas-bot, AnomieBOT,Arthur-Bot, Xqbot, GrouchoBot, SassoBot, AllCluesKey, MondalorBot, ZéroBot, BG19bot, Mark viking and Anonymous: 21

• Limit point compact Source: http://en.wikipedia.org/wiki/Limit_point_compact?oldid=544713924 Contributors: Paul August, Linas,Algebraist, Silly rabbit, Gandalfxviv, Myasuda, Perturbationist, Addbot, Topology Expert, Semistablesystem, 777sms, WikitanvirBot,Paolo Lipparini and Anonymous: 3

• Lindelöf space Source: http://en.wikipedia.org/wiki/Lindel%C3%B6f_space?oldid=643741409 Contributors: Zundark, Michael Hardy,Dominus, Loren Rosen, Revolver, Lumos3, Robinh, Tobias Bergemann, Tosha, Fropuff, Paul August, BenjBot, Vipul, Burn, Linas, R.e.b.,YurikBot, Hairy Dude, Hennobrandsma,Mysid, Bota47, Kompik, SmackBot, OdMishehu, Silly rabbit, Vina-iwbot~enwiki, Stotr~enwiki,Cydebot, Nadav1, Sullivan.t.j, David Eppstein, Marcosaedro, Popopp, JackSchmidt, Andrewbt, DragonBot, MystBot, Addbot, Cuaxdon,LaaknorBot, Ginosbot, Yobot, Citation bot, Xqbot, BenzolBot, Citation bot 1, RedBot, EmausBot, WikitanvirBot, Slawekb, ZéroBot,CitationCleanerBot, Brad7777, YFdyh-bot, Hamoudafg, K9re11, Forgetfulfunctor00, Zdell271 and Anonymous: 15

• Locally compact space Source: http://en.wikipedia.org/wiki/Locally_compact_space?oldid=666366977 Contributors: AxelBoldt, Zun-dark, Toby~enwiki, Toby Bartels, Michael Hardy, Charles Matthews, Jitse Niesen, Lumos3, Shantavira, Robbot, MathMartin, Tosha,Giftlite, Dbenbenn, K igor k, Lupin, Fropuff, Paul August, Vipul, Oleg Alexandrov, BD2412, Jshadias, FlaBot, Mathbot, John Z, Yurik-Bot, Hairy Dude, SmackBot, Andy M. Wang, Bluebot, Silly rabbit, A Geek Tragedy, HLwiKi, Gala.martin, Danpovey, Gandalfxviv,Mets501, Myasuda, Mct mht, Equendil, Cydebot, Sagaciousuk, Headbomb, GurchBot, R'n'B, Ale2006, Marcosaedro, Wikimorphism,JackSchmidt, Addbot, Ginosbot, Luckas-bot, Amirobot, Citation bot, Xqbot, מדר ,יובל Citation bot 1, I dream of horses, EmausBot,PatrickR2, ChrisGualtieri, Gmkwo, Hymath, K9re11, Cohomology84 and Anonymous: 28

• Locally finite Source: http://en.wikipedia.org/wiki/Locally_finite?oldid=572023546 Contributors: Zundark, Revolver, Melchoir, CBM,Addbot, Yaddie, ZéroBot and Anonymous: 1

• Locally finite collection Source: http://en.wikipedia.org/wiki/Locally_finite_collection?oldid=661263548Contributors: Zundark,MichaelHardy, Jitse Niesen, Tobias Bergemann, Giftlite, OdedSchramm, RussBot, Archelon, SmackBot, Silly rabbit, Lambiam, Konradek, R'n'B,JackSchmidt, Sun Creator, Addbot, Topology Expert, Yobot, Ht686rg90, Point-set topologist, ZéroBot, PatrickR2, Helpful Pixie Bot,Pratyush Sarkar and Anonymous: 6

• Locally finite space Source: http://en.wikipedia.org/wiki/Locally_finite_space?oldid=572144183 Contributors: Melchoir, Silly rabbitand David Eppstein

• Mesocompact space Source: http://en.wikipedia.org/wiki/Mesocompact_space?oldid=544373876 Contributors: Zundark, Cyde, Vipul,Linas, SmackBot, Silly rabbit, Cydebot, Thijs!bot, Addbot and Anonymous: 1

• Metacompact space Source: http://en.wikipedia.org/wiki/Metacompact_space?oldid=662330601 Contributors: Zundark, Tobias Berge-mann, Fropuff, Vipul, Linas, Awis, Rjwilmsi, Algebraist, Silly rabbit, CBM, Cydebot, Ntsimp, Vanish2, JackSchmidt, Addbot, TopologyExpert, Luckas-bot, Citation bot 1, Trappist the monk, Schojoha and Anonymous: 2

• Orthocompact space Source: http://en.wikipedia.org/wiki/Orthocompact_space?oldid=544373090Contributors: Zundark, Fropuff, Vipul,Linas, Rjwilmsi, Hennobrandsma, SmackBot, Silly rabbit, CMG, Cydebot, Vanish2, Addbot, Amirobot, Andytoh and Helpful Pixie Bot

• Paracompact space Source: http://en.wikipedia.org/wiki/Paracompact_space?oldid=660911615Contributors: AxelBoldt, Zundark, Toby~enwiki,TobyBartels, Michael Hardy, Revolver, CharlesMatthews, Dfeuer, Dysprosia, JitseNiesen, Fibonacci, Tobias Bergemann,Weialawaga~enwiki,Tosha, Giftlite, Lethe, Fropuff, Brockert, Paul August, Vipul, Kuratowski’s Ghost, Don Reba, ABCD, Oleg Alexandrov, OdedSchramm,Jshadias, Salix alba, R.e.b., Mathbot, Trovatore, Hennobrandsma, SmackBot, Navilluskram, Silly rabbit, Dreadstar, Turms, Mets501,Stotr~enwiki, CBM, Cydebot, Thijs!bot, Headbomb, Vanish2, Jakob.scholbach, David Eppstein, Policron, Hqb, Marcosaedro, Mr.Axiom,MikeRumex, Quietbritishjim,Mrw7,Megaloxantha, Grubb257, AlexeyMuranov, Protony~enwiki, Addbot, Topology Expert, Dingo1729,,ماني Yobot, Kilom691, Hank hu, Citation bot, Txebixev, Howard McCay, LucienBOT, Lost-n-translation, Jonesey95, DixonDBot,Dinamik-bot, EmausBot, Slawekb, Drusus 0, Anselrill, TobiTobsensWiki, Brad7777, Dexbot, Mark viking, Mat.wyszynski and Anony-mous: 29

• Pseudocompact space Source: http://en.wikipedia.org/wiki/Pseudocompact_space?oldid=622051894 Contributors: Fropuff, Elroch,Vipul, Oleg Alexandrov, Linas, Rjwilmsi, Algebraist, Bluebot, Silly rabbit, Vaughan Pratt, Jac16888, Cydebot, Sullivan.t.j, David Epp-stein, SieBot, Addbot, Topology Expert, Luckas-bot, Yobot, Citation bot 1, RjwilmsiBot, Schojoha and Anonymous: 3

• Realcompact space Source: http://en.wikipedia.org/wiki/Realcompact_space?oldid=625743690 Contributors: D6, Vipul, R.e.b., Sillyrabbit, CBM, Myasuda, Jac16888, Cydebot, Headbomb, Agricola44, Jaan Vajakas, Addbot, Erik9bot, Marcus0107, Schojoha, Markviking, Chen10k2 and Anonymous: 3

• Relatively compact subspace Source: http://en.wikipedia.org/wiki/Relatively_compact_subspace?oldid=543779380 Contributors: TobyBartels, Michael Hardy, Revolver, Charles Matthews, ElBenevolente, Tosha, Aylex~enwiki, Linas, Algebraist, Zwobot, SmackBot, Sillyrabbit, DHN-bot~enwiki, Konradek, Sullivan.t.j, Kakila, Addbot, Erik9bot, EmausBot, ZéroBot, CocuBot, PatrickR2 and Anonymous:7

• Sequentially compact space Source: http://en.wikipedia.org/wiki/Sequentially_compact_space?oldid=656966423Contributors: MichaelShulman, TakuyaMurata, Tobias Bergemann, Tosha, Smjg, Fropuff, Paul August, StradivariusTV, Algebraist, RonnieBrown, Silly rab-bit, Obueno~enwiki, Jay Gatsby, Plclark, Talan Gwynek, Addbot, Topology Expert, Citation bot, Edsegal, J04n, Point-set topologist,FrescoBot, EmausBot, WikitanvirBot, ZéroBot, Chharvey, Helpful Pixie Bot, Paolo Lipparini, Mark viking and Anonymous: 13

35.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 103

• Strictly singular operator Source: http://en.wikipedia.org/wiki/Strictly_singular_operator?oldid=663253915Contributors: Michael Hardy,Linas, Jmath666, Addbot, AnomieBOT, John of Reading, Mgkrupa and Anonymous: 4

• Subset Source: http://en.wikipedia.org/wiki/Subset?oldid=658760669 Contributors: Damian Yerrick, AxelBoldt, Youssefsan, XJaM,Toby Bartels, StefanRybo~enwiki, Edward, Patrick, TeunSpaans, Michael Hardy, Wshun, Booyabazooka, Ellywa, Oddegg, Andres,Charles Matthews, Timwi, Hyacinth, Finlay McWalter, Robbot, Romanm, Bkell, 75th Trombone, Tobias Bergemann, Tosha, Giftlite,Fropuff, Waltpohl, Macrakis, Tyler McHenry, SatyrEyes, Rgrg, Vivacissamamente, Mormegil, EugeneZelenko, Noisy, Deh, Paul Au-gust, Engmark, Spoon!, SpeedyGonsales, Obradovic Goran, Nsaa, Jumbuck, Raboof, ABCD, Sligocki, Mac Davis, Aquae, LFaraone,Chamaeleon, Firsfron, Isnow, Salix alba, VKokielov, Mathbot, Harmil, BMF81, Chobot, Roboto de Ajvol, YurikBot, Alpt, Dmharvey,KSmrq, NawlinWiki, Trovatore, Nick, Szhaider, Wasseralm, Sardanaphalus, Jacek Kendysz, BiT, Gilliam, Buck Mulligan, SMP, Or-angeDog, Bob K, Dreadstar, Bjankuloski06en~enwiki, Loadmaster, Vedexent, Amitch, Madmath789, Newone, CBM, Jokes Free4Me,345Kai, SuperMidget, Gregbard, WillowW, MC10, Thijs!bot, Headbomb, Marek69, RobHar, WikiSlasher, Salgueiro~enwiki, JAnDbot,.anacondabot, Pixel ;-), Pawl Kennedy, Emw, ANONYMOUS COWARD0xC0DE, RaitisMath, JCraw, Tgeairn, Ttwo, Maurice Car-bonaro, Acalamari, Gombang, NewEnglandYankee, Liatd41, VolkovBot, CSumit, Deleet, Rei-bot, AnonymousDissident, James.Spudeman,PaulTanenbaum, InformationSpace, Falcon8765, AlleborgoBot, P3d4nt, NHRHS2010, Garde, Paolo.dL, OKBot, Brennie8, Jons63,Loren.wilton, ClueBot, GorillaWarfare, PipepBot, The Thing That Should Not Be, DragonBot, Watchduck, Hans Adler, Computer97,Noosentaal, Versus22, PCHS-NJROTC, Andrew.Flock, Reverb123, Addbot, , Fyrael, PranksterTurtle, Numbo3-bot, Zorrobot, Jar-ble, JakobVoss, Luckas-bot, Yobot, Synchronism, AnomieBOT, Jim1138, Materialscientist, Citation bot, Martnym, NFD9001, Char-vest, 78.26, XQYZ, Egmontbot, Rapsar, HRoestBot, Suffusion of Yellow, Agent Smith (The Matrix), RenamedUser01302013, ZéroBot,Alexey.kudinkin, Chharvey, Quondum, Chewings72, 28bot, ClueBot NG, Wcherowi, Matthiaspaul, Bethre, Mesoderm, O.Koslowski,AwamerT, Minsbot, Pratyya Ghosh, YFdyh-bot, Ldfleur, ChalkboardCowboy, Saehry, Stephan Kulla, , Ilya23Ezhov, Sandshark23,Quenhitran, Neemasri, Prince Gull, Maranuel123, Alterseemann, Rahulmr.17 and Anonymous: 179

• Supercompact space Source: http://en.wikipedia.org/wiki/Supercompact_space?oldid=655715022Contributors: Zundark,Michael Hardy,Rich Farmbrough, Vipul, C S, RJFJR, OdedSchramm, Rjwilmsi, SmackBot, Silly rabbit, Baa, TenPoundHammer, Lambiam, Olaf Davis,Cydebot, David Eppstein, TallNapoleon, Anturiaethwr, Topology Expert, Citation bot, Citation bot 1, Trappist the monk, Suslindisam-biguator, Helpful Pixie Bot and Anonymous: 1

• Topological space Source: http://en.wikipedia.org/wiki/Topological_space?oldid=662870663 Contributors: AxelBoldt, Zundark, XJaM,Toby Bartels, Olivier, Patrick, Michael Hardy, Wshun, Kku, Dineshjk, Karada, Hashar, Zhaoway~enwiki, Revolver, Charles Matthews,Dcoetzee, Dysprosia, Kbk, Taxman, Phys, Robbot, Nizmogtr, Fredrik, Saaska, MathMartin, P0lyglut, Tobias Bergemann, Giftlite, GeneWard Smith, Lethe, Fropuff, Dratman, DefLog~enwiki, Rhobite, Luqui, Paul August, Dolda2000, Elwikipedista~enwiki, Tompw, Aude,SgtThroat, Tsirel, Marc van Woerkom, Varuna, Kuratowski’s Ghost, Msh210, Keenan Pepper, Danog, Sligocki, Spambit, Oleg Alexan-drov, Woohookitty, Graham87, BD2412, Grammarbot, FlaBot, Sunayana, Tillmo, Chobot, Algebraist, YurikBot, Wavelength, HairyDude, NawlinWiki, Rick Norwood, Bota47, Stefan Udrea, Hirak 99, Arthur Rubin, Lendu, JoanneB, Eaefremov, RonnieBrown, Sar-danaphalus, SmackBot, Maksim-e~enwiki, Sciyoshi~enwiki, DHN-bot~enwiki, Tsca.bot, Tschwenn, LkNsngth, Vriullop, Arialblack, Iri-descent, Devourer09,Mattbr, AndrewDelong, Kupirijo, Roccorossi, Xantharius, Thijs!bot, Konradek, Odoncaoa, Escarbot, Salgueiro~enwiki,JAnDbot, YK Times, Jakob.scholbach, Bbi5291, Wdevauld, J.delanoy, Pharaoh of the Wizards, Maurice Carbonaro, The Mudge, Jma-jeremy, Policron, TXiKiBoT,AnonymousDissident, Plclark, AaronRotenberg, Jesin, Arcfrk, SieBot, MiNombreDeGuerra, JerroldPease-Atlanta, JackSchmidt, Failure.exe, Egmontaz, Palnot, SilvonenBot, Addbot, CarsracBot, AnnaFrance, ChenzwBot, Luckas-bot, Yobot,SwisterTwister, AnomieBOT, Ciphers, Materialscientist, Citation bot, DannyAsher, FlordiaSunshine342, J04n, Point-set topologist, Fres-coBot, Jschnur, Jeroen De Dauw, TobeBot, Seahorseruler, Skakkle, Cstanford.math, ZéroBot, Chharvey, Wikfr, Orange Suede Sofa,Liuthar, ClueBot NG, Wcherowi, Mesoderm, Vinícius Machado Vogt, Helpful Pixie Bot, Gaurav Nirala, Tom.hirschowitz, Pacerier,Cpatra1984, Brad7777, Minsbot, LoganFromSA, MikeHaskel, Acer4666, Freeze S, Mark viking, Epicgenius, Kurt Artindagi, Improba-ble keeler, Amonk1962, KasparBot and Anonymous: 108

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104 CHAPTER 35. Σ-COMPACT SPACE

Zorrobot, TeH nOmInAtOr, Jarble, Sammtamm, Legobot, Cote d'Azur, Luckas-bot, 2D, Deputyduck, AnomieBOT, 1exec1, Jack-ieBot, AdjustShift, Materialscientist, Citation bot, Xqbot, Ekwos, J04n, GrouchoBot, Point-set topologist, RibotBOT, Charvest, Con-traverse, Divisbyzero, Orhanghazi, VI, Anilkumarphysics, Commit charge, Pinethicket, Tom.Reding, CrowzRSA, PoincaresChild, To-beBot, Jws401, Enthdegree, Integrals4life, Unbitwise, Jesse V., EmausBot, Fly by Night, Slawekb, ZéroBot, Chimpdmunk, The Nut,Caspertheghost, QEDK, Staszek Lem, Lorem Ip, ProteoPhenom, Anita5192, ResearchRave, ClueBot NG, Wcherowi, O.Koslowski,Widr, Helpful Pixie Bot, Daheadhunter, BG19bot, TCN7JM, Bigdon128, Wimvdam, Brad7777, Charismaa, Waleed.598, Sboosali, JY-Bot, MrBubbleFace, Dexbot, Paulo Henrique Macedo, King jakob c, Brirush, Mark viking, Ayesh2788, I am One of Many, TJLaher123,SakeUPenn, K401sTL3, Lizia7, Sesamo12, Btomoiaga, Chuluojun, Je.est.un.autre, Betapictoris, SoSivr, Jainmskip, Hriton, KasparBotand Anonymous: 348

• Totally bounded space Source: http://en.wikipedia.org/wiki/Totally_bounded_space?oldid=652130953 Contributors: Awaterl, TobyBartels, MathMartin, Tobias Bergemann, Fropuff, Paul August, Oleg Alexandrov, Simetrical, Koroner, Zwobot, Kompik, Silly rabbit,A Geek Tragedy, Dreadstar, JHunterJ, Thijs!bot, Salgueiro~enwiki, GromXXVII, McM.bot, Megaloxantha, M.landgren, Niceguyedc,Luca Antonelli, Addbot, Luckas-bot, Yobot, Citation bot, Lost-n-translation, DrilBot, Junior Wrangler, Espresso-hound, WikitanvirBot,DPL bot, Noix07 and Anonymous: 24

• Σ-compact space Source: http://en.wikipedia.org/wiki/%CE%A3-compact_space?oldid=646657935Contributors: BryanDerksen, Zun-dark, Michael Hardy, TakuyaMurata, Giftlite, Fropuff, Guanaco, Paul August, Teorth, Oleg Alexandrov, Bgwhite, Trovatore, Silly rabbit,PamD, VolkovBot, Plclark, Blurpeace, AlleborgoBot, JackSchmidt, Tomas e, Addbot, Topology Expert, Yobot, Erik9bot, SławomirBiały, BenzolBot, Stj6, RjwilmsiBot, EmausBot, WikitanvirBot, Mgkrupa, Reznov-kuratow and Anonymous: 6

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