Locally Finite Collection

Locally nite collectionFrom Wikipedia, the free encyclopedia

Contents

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

2 Closure (topology) 22.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.3.1 Open cover denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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4.3.2 Equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Denition (topological spaces) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Denition (normed spaces) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6 Cover (topology) 206.1 Cover in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.2 Renement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.2 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

14 Lebesgue covering dimension 4114.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Lindelf space 4516.1 Properties of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.2 Properties of strongly Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.3 Product of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.4 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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

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

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

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

18 Locally nite 51

19 Locally nite 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 nite collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5319.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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20 Mesocompact space 5420.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5420.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

21 Metacompact space 5521.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5521.2 Covering dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5521.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5521.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

22 Orthocompact space 5722.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

23 Paracompact space 5823.1 Paracompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5823.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5823.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5923.4 Paracompact Hausdor Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

23.4.1 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6023.5 Relationship with compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

23.5.1 Comparison of properties with compactness . . . . . . . . . . . . . . . . . . . . . . . . . 6123.6 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

23.6.1 Denition of relevant terms for the variations . . . . . . . . . . . . . . . . . . . . . . . . . 6223.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6223.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6223.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6323.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

24 Pseudocompact space 6424.1 Properties related to pseudocompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6424.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6424.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

25 Realcompact space 6625.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6625.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6625.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

26 Relatively compact subspace 6826.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6826.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

27 Sequentially compact space 6927.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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27.2 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6927.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6927.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6927.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

28 Strictly singular operator 7128.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

29 Subset 7229.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7329.2 and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7329.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7329.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7429.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7429.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7429.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

30 Supercompact space 7630.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7630.2 Some Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7630.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

31 Topological space 7831.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

31.1.1 Neighbourhoods denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7831.1.2 Open sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7931.1.3 Closed sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8031.1.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

31.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8031.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8031.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8131.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8231.6 Classication of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8231.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8231.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8231.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8231.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8331.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8331.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8331.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

32 Topology 8532.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

CONTENTS vii

32.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8732.3 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

32.3.1 Topologies on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8932.3.2 Continuous functions and homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 9032.3.3 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

32.4 Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9032.4.1 General topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9032.4.2 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9132.4.3 Dierential topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9132.4.4 Geometric topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9132.4.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

32.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9232.5.1 Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9232.5.2 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9232.5.3 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9232.5.4 Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

32.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9232.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9332.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9432.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

33 Totally bounded space 9533.1 Denition for a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9533.2 Denitions in other contexts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9533.3 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9633.4 Relationships with compactness and completeness . . . . . . . . . . . . . . . . . . . . . . . . . . 9633.5 Use of the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9733.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9733.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9733.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

34 -compact space 9834.1 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9834.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9834.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9934.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9934.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 100

34.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10034.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10334.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Chapter 1

a-paracompact space

In mathematics, in the eld of topology, a topological space is said to be a-paracompact if every open cover of thespace has a locally nite renement. In contrast to the denition of paracompactness, the renement 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 dened as the union of S and its boundary. Intuitively, these are all the points in S and nearS. 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 Denitions

2.1.1 Point of closureFor 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 denition 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) := inf{d(x, s) : s inS} = 0.This denition 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 denition does not depend upon whether neighbourhoods are required to beopen.

2.1.2 Limit pointThe denition of a point of closure is closely related to the denition of a limit point. The dierence between thetwo denitions is subtle but important namely, in the denition 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 setSee 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 denition of the topological closure, which still makesense when applied to other types of closures (see below).[5]

In a rst-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 "lter".Note that these properties are also satised if closure, superset, intersection, contains/containing, smallestand closed are replaced by interior, subset, union, contained in, largest, and open. For more on thismatter, see closure operator below.

2.2 ExamplesConsider 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 nite 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

p2:

2.3 Closure operatorSee also: Closure operator

A closure operator on a set X is a mapping of the power set of X, P(X) , into itself which satises 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 dening 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-theoreticdierence.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 closuresThe 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 nitely 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 innitely 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 interpretationOne may elegantly dene 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 xed subset A X can be identiedwith 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 nA) 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 denition and a few properties of the above categories. More-over, this denition 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 denition.

[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 nite rank is a compact operator; indeed, the class of compact operators is a naturalgeneralisation of the class of nite-rank operators in an innite-dimensional setting. When Y is a Hilbert space, itis true that any compact operator is a limit of nite-rank operators, so that the class of compact operators can bedened alternatively as the closure in the operator norm of the nite-rank operators. Whether this was true in generalfor Banach spaces (the approximation property) was an unsolved question for many years; in the end Per Eno 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 nite-rankoperators is basic in the numerical solution of such equations. The abstract idea of Fredholm operator is derived fromthis connection.

3.1 Equivalent formulationsA 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)n2N from the unit ball in X, the sequence (Txn)n2N 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 propertiesIn 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 nite 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 nite 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 isnite-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 (Schauders theorem).

3.3 Origins in integral equation theoryA crucial property of compact operators is the Fredholm alternative, which asserts that the existence of solution oflinear equations of the form(K + 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 nite 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 innite-dimensional Banach space has spectrum that is either anite subset of C which includes 0, or the spectrum is a countably innite 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 nite multiplicities(so that K I has a nite-dimensional kernel for all complex 0).An important example of a compact operator is compact embedding of Sobolev spaces, which, along with the Grdinginequality and the LaxMilgram 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 innitely 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 innite-dimensional Hilbert space form a maximal ideal, so thequotient algebra, known as the Calkin algebra, is simple.

3.4 Compact operator on Hilbert spacesMain article: Compact operator on Hilbert space

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

T : H ! His said to be compact if it can be written in the form

T =1Xn=1

nhfn; ign ;

8 CHAPTER 3. COMPACT OPERATOR

where f1; f2; : : : and g1; g2; : : : are (not necessarily complete) orthonormal sets, and 1; 2; : : : 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 N+k = 0 for some N 2 N; and every k = 1; 2; : : : ,then the operator has nite rank, i.e, a nite-dimensional range and can be written as

T =NXn=1

nhfn; ign :

The bracket h; i 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 operatorsLet 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 reexive Banach space, then everycompletely continuous operator T : X Y is compact.

3.6 Examples Every nite rank operator is compact.

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

For some xed g C([0, 1]; R), dene the linear operator T from C([0, 1]; R) to C([0, 1]; R) by

(Tf)(x) =

Z x0

f(t)g(t) dt:

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

More generally, if is any domain in Rn and the integral kernel k : R is a HilbertSchmidt kernel,then the operator T on L2(; R) dened by

(Tf)(x) =

Z

k(x; y)f(y) dy

is a compact operator.

By Rieszs lemma, the identity operator is a compact operator if and only if the space is nite-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 dierential 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 rst-order logic, see Compactness theorem.In mathematics, and more specically in general topology, compactness is a property that generalizes the notion of

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

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

10

4.1. HISTORICAL DEVELOPMENT 11

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

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

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

12 CHAPTER 4. COMPACT SPACE

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

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

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

fUg2Aof open subsets of X such that

X =[2A

U;

there is a nite subset J of A such that

X =[i2J

Ui:

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

4.3. DEFINITIONS 13

4.3.2 Equivalent denitions

Assuming the axiom of choice, the following are equivalent:

1. A topological space X is compact.

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

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

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

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

6. Every lter on X has a convergent renement.

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

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

Euclidean space

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

Metric spaces

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

1. (X,d) is compact.

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

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

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

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

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

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

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

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


Characterization by continuous functions

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

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

Hyperreal denition

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

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

fUg2Aof open subsets of X such that

K [2A

U;

there is a nite subset J of A such that

K [i2J

Ui:

4.4 Properties of compact spaces

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

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

4.5. EXAMPLES 15

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

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

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

Any space carrying the conite topology is compact. Any locally compact Hausdor space can be turned into a compact space by adding a single point to it, bymeans of Alexandro one-point compactication. The one-point compactication of R is homeomorphic tothe circle S1; the one-point compactication of R2 is homeomorphic to the sphere S2. Using the one-pointcompactication, one can also easily construct compact spaces which are not Hausdor, by starting with anon-Hausdor space.

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

R, carrying the lower limit topology, satises the property that no uncountable set is compact. In the cocountable topology on an uncountable set, no innite set is compact. Like the previous example, thespace as a whole is not locally compact but is still Lindelf.

The closed unit interval [0,1] is compact. This follows from the HeineBorel theorem. The open interval (0,1)is not compact: the open cover

1

n; 1 1

n

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

1

1n

and

1

+

1

n; 1

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

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

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

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


The Cantor set is compact. In fact, every compact metric space is a continuous image of the Cantor set. Consider the set K of all functions f : R [0,1] from the real number line to the closed unit interval, and denea topology on K so that a sequence ffng in K converges towards f 2 K if and only if ffn(x)g convergestowards f(x) for all real numbers x. There is only one such topology; it is called the topology of pointwiseconvergence or the product topology. Then K is a compact topological space; this follows from the Tychonotheorem.

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

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

jf(x) g(x)j:

Then by ArzelAscoli theorem the space K is compact.

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

4.5.1 Algebraic examples Compact groups such as an orthogonal group are compact, while groups such as a general linear group are not. Since the p-adic integers are homeomorphic to the Cantor set, they form a compact set. The spectrum of any commutative ring with the Zariski topology (that is, the set of all prime ideals) is compact,but never Hausdor (except in trivial cases). In algebraic geometry, such topological spaces are examples ofquasi-compact schemes, quasi referring to the non-Hausdor nature of the topology.

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

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

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

4.7. NOTES 17

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

[2] Kline 1972, Chapter 46, 2

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

[4] (Kelley 1955, p. 163)

[5] Arkhangelskii & Fedorchuk 1990, Theorem 5.3.7

[6] Willard 1970 Theorem 30.7.

[7] Gillman & Jerison 1976, 5.6

[8] Robinson, Theorem 4.1.13

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

[10] Arkhangelskii & Fedorchuk 1990, Corollary 5.2.1

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

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

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

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

Arkhangelskii, A.V. (2001), Compact space, 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 gewhren, wenigstens eine reele Wurzel der Gleichung liege, Wilhelm Engelmann (Purelyanalytic proof of the theorem that between any two values which give results of opposite sign, there lies at leastone real root of the equation).

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

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

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

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

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

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

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


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

Lebesgue, Henri (1904), Leons sur l'intgration et la recherche des fonctions primitives, Gauthier-Villars. Robinson, Abraham (1996), Non-standard analysis, Princeton University Press, ISBN 978-0-691-04490-3,MR 0205854.

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

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

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

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

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

Chapter 5

Compactly embedded

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

5.1 Denition (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 Denition (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 denition 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 dierential equations. Providence, RI: American Mathematical Society.ISBN 0-8218-0772-2..

Renardy, M., & Rogers, R. C. (1992). An Introduction to Partial Dierential 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 = fU : 2 Ag

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

X [2A

U:

6.1 Cover in topologyCovers 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 U of X whose union is the whole space X. In this case we say that C covers X, or that the setsU 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 [2A

U

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 U is contained in T, where T is thetopology on X).A cover of X is said to be locally nite if every point of X has a neighborhood which intersects only nitely many setsin the cover. Formally, C = {U} is locally nite if for any x X, there exists some neighborhood N(x) of x suchthat the set

f 2 A : U \N(x) 6= ?g

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

6.2 RenementA renement 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 = V2B

is a renement of

U2A when 8 9 V UIn other words, there is a renement map : B ! A satisfying V U() for every 2 B . This map is used,for instance, in the ech cohomology of X.[1]

Every subcover is also a renement, 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 renement is made from any sets that are subsets of the sets in thecover.The renement relation is a preorder on the set of covers of X.Generally speaking, a renement of a given structure is another that in some sense contains it. Examples are to befound when partitioning an interval (one renement of a0 < a1 < ::: < an being a0 < b0 < a1 < a2 < ::: < an 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 Hausdor, topological vector space that is innite-dimensional, such as an innite-dimensionalHilbert space.

The rst 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 specic, a Hausdor topological vector space is locally compact if and only if it isnite-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-Hausdor examples The one-point compactication 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 innite 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 PropertiesEvery locally compact preregular space is, in fact, completely regular. It follows that every locally compact Hausdorspace is a Tychono 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 Tychonospaces are usually just referred to as locally compact Hausdor spaces.Every locally compact Hausdor 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 Hausdor space Y is locally compact if and only if X can be written as the set-theoretic dierence of two closed subsets of Y. As a corollary, a dense subspace X of a locally compact Hausdorspace Y is locally compact if and only if X is an open subset of Y. Furthermore, if a subspace X of any Hausdorspace Y is locally compact, then X still must be the dierence of two closed subsets of Y, although the converseneedn't hold in this case.Quotient spaces of locally compact Hausdor spaces are compactly generated. Conversely, every compactly gener-ated Hausdor space is a quotient of some locally compact Hausdor space.For locally compact spaces local uniform convergence is the same as compact convergence.

17.3.1 The point at innitySince every locally compact Hausdor space X is Tychono, it can be embedded in a compact Hausdor space b(X)using the Stoneech compactication. But in fact, there is a simpler method available in the locally compact case;the one-point compactication will embed X in a compact Hausdor space a(X) with just one extra point. (The one-point compactication can be applied to other spaces, but a(X) will be Hausdor if and only if X is locally compactand Hausdor.) The locally compact Hausdor spaces can thus be characterised as the open subsets of compactHausdor spaces.Intuitively, the extra point in a(X) can be thought of as a point at innity. The point at innity should be thought ofas lying outside every compact subset of X. Many intuitive notions about tendency towards innity can be formulatedin locally compact Hausdor spaces using this idea. For example, a continuous real or complex valued function fwith domain X is said to vanish at innity 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 denition makes sense for any topological space X. IfX is locally compact and Hausdor, such functions are precisely those extendable to a continuous function g on itsone-point compactication a(X) = X {} where g() = 0.The set C0(X) of all continuous complex-valued functions that vanish at innity is a C* algebra. In fact, everycommutative C* algebra is isomorphic to C0(X) for some unique (up to homeomorphism) locally compact Hausdorspace X. More precisely, the categories of locally compact Hausdor spaces and of commutative C* algebras aredual; this is shown using the Gelfand representation. Forming the one-point compactication a(X) of X correspondsunder this duality to adjoining an identity element to C0(X).

17.3.2 Locally compact groupsThe notion of local compactness is important in the study of topological groups mainly because every Hausdorlocally compact group G carries natural measures called the Haar measures which allow one to integrate measurablefunctions dened 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 denes a self-duality of the category of locally compact abelian groups. The study oflocally compact abelian groups is the foundation of harmonic analysis, a eld 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 nite

The term locally nite has a number of dierent meanings in mathematics:

Locally nite collection of sets in a topological space Locally nite group Locally nite measure Locally nite operator in linear algebra Locally nite poset Locally nite space, a topological space in which each point has a nite neighborhood Locally nite variety in the sense of universal algebra

51

Chapter 19

Locally nite collection

In the mathematical eld of topology, local niteness 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 nite, if each point in the space has a neighbourhoodthat intersects only nitely many of the sets in the collection.Note that the term locally nite has dierent meanings in other mathematical elds.

19.1 Examples and propertiesA nite collection of subsets of a topological space is locally nite. Innite collections can also be locally nite: 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 nite, 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 nite, the collection of all closures of these sets is also locally nite. 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 nite complement topology on R the collection of all open sets is not locally nite, butthe collection of all closures of these sets is locally nite (since the only closures are R and the empty set).

19.1.1 Compact spacesNo innite collection of a compact space can be locally nite. Indeed, let {Ga} be an innite family of subsets ofa space and suppose this collection is locally nite. For each point x of this space, choose a neighbourhood Ux thatintersects the collection {Ga} at only nitely 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 nite subcover, Ua1 ...... Uan. Since each Uai intersects {Ga} for only nitely many values ofa, the union of all such Uai intersects the collection {Ga} for only nitely many values of a. It follows that X (thewhole space!) intersects the collection {Ga} at only nitely many values of a, contradicting the innite cardinality ofthe collection {Ga}.A topological space in which every open cover admits a locally nite open renement is called paracompact. Everylocally nite collection of subsets of a topological space X is also point-nite. A topological space in which everyopen cover admits a point-nite open renement is called metacompact.

19.1.2 Second countable spacesNo uncountable cover of a Lindelf space can be locally nite, by essentially the same argument as in the case ofcompact spaces. In particular, no uncountable cover of a second-countable space is locally nite.

52

19.2. CLOSED SETS 53

19.2 Closed setsIt is clear from the denition of a topology that a nite union of closed sets is closed. One can readily give an exampleof an innite union of closed sets that is not closed. However, if we consider a locally nite 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 nite collection of closedsets, we merely choose a neighbourhoodV of x that intersects this collection at only nitely many of these sets. Denea 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 nite collectionsA collection in a space is countably locally nite (or -locally nite) if it is the union of a countable family of locallynite collections of subsets of X. Countable local niteness is a key hypothesis in the NagataSmirnov metrizationtheorem, which states that a topological space is metrizable if and only if it is regular and has a countably locally nitebasis.

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

Chapter 20

Mesocompact space

In mathematics, in the eld of general topology, a topological space is said to bemesocompact if every open cover hasa compact-nite open renement.[1] That is, given any open cover, we can nd an open renement with the propertythat every compact set meets only nitely many members of the renement.[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 nite cover is automatically compact-nite.

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

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

[2] Pearl, p23

20.2 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

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Chapter 21

Metacompact space

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

21.1 PropertiesThe 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 Dieudonn 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 Tychono space X to be compact it is necessary and sucient that X be metacompact andpseudocompact (see Watson).

21.2 Covering dimensionA topological space X is said to be of covering dimension n if every open cover of X has a point nite open renementsuch that no point of X is included in more than n + 1 sets in the renement and if n is the minimum value for whichthis is true. If no such minimal n exists, the space is said to be of innite covering dimension.

21.3 See also Compact space Paracompact space Normal space Realcompact space Pseudocompact space

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56 CHAPTER 21. METACOMPACT SPACE

Mesocompact space Tychono space Glossary of topology

21.4 References Watson, W. Stephen (1981). Pseudocompact metacompact spaces are compact. Proc. Amer. Math. Soc.81: 151152. 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 22

Orthocompact space

In mathematics, in the eld of general topology, a topological space is said to be orthocompact if every open coverhas an interior preserving open renement. That is, given an open cover of the topological space, there is a renementwhich is also an open cover, with the further property that at any point, the intersection of all open sets in the renementcontaining that point, is also open.If the number of open sets containing the point is nite, then their intersectio

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