Connected spaceFrom Wikipedia, the free encyclopedia

Contents

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

2 Alexandrov topology 22.1 Characterizations of Alexandrov topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Duality with preordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 The Alexandrov topology on a preordered set . . . . . . . . . . . . . . . . . . . . . . . . 32.2.2 The specialization preorder on a topological space . . . . . . . . . . . . . . . . . . . . . . 32.2.3 Equivalence between preorders and Alexandrov topologies . . . . . . . . . . . . . . . . . 32.2.4 Equivalence between monotony and continuity . . . . . . . . . . . . . . . . . . . . . . . . 42.2.5 Category theoretic description of the duality . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.6 Relationship to the construction of modal algebras from modal frames . . . . . . . . . . . 5

2.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Annulus (mathematics) 73.1 Complex structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Baire space 104.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2.1 Modern denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2.2 Historical denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4 Baire category theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.8 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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5 Collectionwise Hausdor space 135.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Collectionwise normal space 146.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

7 Compact space 157.1 Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.2 Basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.3 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

7.3.1 Open cover denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.3.2 Equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.3.3 Compactness of subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7.4 Properties of compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.4.1 Functions and compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.4.2 Compact spaces and set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.4.3 Ordered compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.5.1 Algebraic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8 Connected space 248.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

8.1.1 Connected components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.1.2 Disconnected spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

8.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.3 Path connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.4 Arc connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.5 Local connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.6 Set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.7 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.8 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.9 Stronger forms of connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

8.11.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.11.2 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

9 Contractible space 329.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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9.2 Locally contractible spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.3 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

10 Countably compact space 3410.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

11 Disjoint sets 3511.1 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.3 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.4 Disjoint unions and partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

12 Disk (mathematics) 3912.1 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

13 Door space 4213.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4213.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

14 Dowker space 4314.1 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4314.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

15 Dyadic space 4415.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

16 End (topology) 4516.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.4 Ends of graphs and groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.5 Ends of a CW complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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17 Extremally disconnected space 4717.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4717.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

18 Feebly compact space 48

19 First-countable space 4919.1 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4919.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4919.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5019.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

20 Glossary of topology 5120.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5220.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5320.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5320.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5520.5 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5520.6 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5520.7 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5620.8 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5620.9 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5720.10K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5720.11L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5820.12M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5820.13N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5920.14O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6020.15P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6020.16Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6120.17R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6220.18S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6220.19T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6320.20U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6420.21W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6520.22Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6520.23References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6520.24External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

21 Grammatical aspect 6721.1 Basic concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

21.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6721.1.2 Modern usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

21.2 Common aspectual distinctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

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21.3 Aspect vs. tense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6821.4 Lexical vs. grammatical aspect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6921.5 Indicating aspect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6921.6 Aspect by language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

21.6.1 English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7021.6.2 German vernacular and colloquial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7121.6.3 Slavic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7221.6.4 Romance languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7221.6.5 Finnic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7321.6.6 Austronesian languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7321.6.7 Creole languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7421.6.8 American Sign Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

21.7 Terms for various aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7521.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7621.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7621.10Other references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7721.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

22 Grammatical mood 7822.1 Realis moods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

22.1.1 Indicative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7822.2 Irrealis moods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

22.2.1 Subjunctive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7922.2.2 Conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8022.2.3 Optative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8022.2.4 Imperative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8022.2.5 Jussive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8122.2.6 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8122.2.7 Inferential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

22.3 Other moods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8122.3.1 Interrogative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8122.3.2 Deity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

22.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8222.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8222.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

23 H-closed space 8323.1 Examples and equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8323.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8323.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

24 HeineBorel theorem 84

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24.1 History and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8424.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8424.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8524.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8624.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8624.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8624.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

25 Hemicompact space 8825.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8825.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8825.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8825.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

26 Hyperconnected space 9026.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9026.2 Hyperconnectedness vs. connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9026.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9026.4 Irreducible components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9126.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9126.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

27 Kolmogorov space 9227.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9227.2 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

27.2.1 Spaces which are not T0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9227.2.2 Spaces which are T0 but not T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

27.3 Operating with T0 spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9327.4 The Kolmogorov quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9327.5 Removing T0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9427.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

28 Limit point compact 9528.1 Properties and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9528.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9528.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

29 Lindelf space 9729.1 Properties of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9729.2 Properties of strongly Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9729.3 Product of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9729.4 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

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29.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9829.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9829.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

30 Locally compact space 9930.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9930.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

30.2.1 Compact Hausdor spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10030.2.2 Locally compact Hausdor spaces that are not compact . . . . . . . . . . . . . . . . . . . 10030.2.3 Hausdor spaces that are not locally compact . . . . . . . . . . . . . . . . . . . . . . . . 10030.2.4 Non-Hausdor examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

30.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10130.3.1 The point at innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10130.3.2 Locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

30.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10230.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

31 Locally connected space 10331.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10431.2 Denitions and rst examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

31.2.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10531.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10531.4 Components and path components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

31.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10631.5 Quasicomponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

31.5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10631.6 More on local connectedness versus weak local connectedness . . . . . . . . . . . . . . . . . . . . 10731.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10731.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10731.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10831.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

32 Locally nite collection 10932.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

32.1.1 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10932.1.2 Second countable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

32.2 Closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11032.3 Countably locally nite collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11032.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

33 Locally nite space 11133.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

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34 Locally Hausdor space 11234.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

35 Locally normal space 11335.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11335.2 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11335.3 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11335.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11335.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

36 Locally regular space 11536.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11536.2 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11536.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11536.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

37 Locally simply connected space 11637.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

38 Luzin space 11838.1 In real analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11838.2 Example of a Luzin set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11838.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

39 Mesocompact space 12039.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12039.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

40 Metacompact space 12140.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12140.2 Covering dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12140.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12140.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

41 Michael selection theorem 12341.1 Other selection theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12341.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

42 Monotonically normal space 12542.1 Equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

42.1.1 Denition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12542.1.2 Denition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12542.1.3 Denition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

42.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

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42.3 Some discussion links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

43 n-connected 12743.1 n-connected space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

43.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12743.2 n-connected map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

43.2.1 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12843.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12943.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

44 Noetherian topological space 13044.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13044.2 Relation to compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13044.3 Noetherian topological spaces from algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . 13044.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13144.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

45 Normal space 13245.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13245.2 Examples of normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13345.3 Examples of non-normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13345.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13445.5 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13445.6 Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13445.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

46 Open set 13546.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13646.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

46.2.1 Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13746.2.2 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13746.2.3 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

46.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13746.4 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13746.5 Notes and cautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

46.5.1 Open is dened relative to a particular topology . . . . . . . . . . . . . . . . . . . . . . 13846.5.2 Open and closed are not mutually exclusive . . . . . . . . . . . . . . . . . . . . . . . . . 138

46.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13846.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13846.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

47 Orthocompact space 14047.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

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48 P-space 14148.1 Generic use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14148.2 P-spaces in the sense of GillmanHenriksen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14148.3 P-spaces in the sense of Morita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14148.4 p-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14148.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14148.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14248.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

49 Paracompact space 14349.1 Paracompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14349.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14349.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14449.4 Paracompact Hausdor Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

49.4.1 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14549.5 Relationship with compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

49.5.1 Comparison of properties with compactness . . . . . . . . . . . . . . . . . . . . . . . . . 14649.6 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

49.6.1 Denition of relevant terms for the variations . . . . . . . . . . . . . . . . . . . . . . . . . 14749.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14749.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14749.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14849.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

50 Paranormal space 14950.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14950.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

51 Path (topology) 15051.1 Homotopy of paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15151.2 Path composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15251.3 Fundamental groupoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15251.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

52 Perfect set 15352.1 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15352.2 Imperfection of a space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15352.3 Closure properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15352.4 Connection with other topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15452.5 Perfect spaces in descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15452.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15452.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

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53 Pluperfect 15553.1 Meaning of the pluperfect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15553.2 Examples from various languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

53.2.1 Greek and Latin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15653.2.2 English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15653.2.3 Other Germanic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15653.2.4 German . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15653.2.5 Dutch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15753.2.6 Romance languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15753.2.7 Slavic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15853.2.8 Other languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

53.3 Table of forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15953.4 Dierent perfect construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15953.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15953.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16053.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

54 Polyadic space 16154.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16154.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16154.3 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16154.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16154.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

54.5.1 Ramseys theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16254.5.2 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

54.6 Generalisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16354.6.1 Centred space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16354.6.2 AD-compact space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16354.6.3 -adic space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16354.6.4 Hyadic space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

54.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16454.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

55 Pseudocompact space 16655.1 Properties related to pseudocompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16655.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16655.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

56 Pseudometric space 16856.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16856.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16856.3 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

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56.4 Metric identication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16956.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16956.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

57 Pseudonormal space 17157.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

58 Realcompact space 17258.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17258.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17258.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

59 Regular space 17459.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17459.2 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17559.3 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17559.4 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17659.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

60 Relatively compact subspace 17760.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17760.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

61 Resolvable space 17861.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17861.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17861.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

62 Rickart space 17962.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

63 Second-countable space 18063.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

63.1.1 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18063.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18163.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

64 Semi-locally simply connected 18264.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18264.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18264.3 Topology of fundamental group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18364.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18364.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

65 Separable space 184

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65.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18465.2 Separability versus second countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18465.3 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18565.4 Constructive mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18565.5 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

65.5.1 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18565.5.2 Non-separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

65.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18665.6.1 Embedding separable metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

65.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

66 Sequential space 18866.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18866.2 Sequential closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18866.3 FrchetUrysohn space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18966.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18966.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18966.6 Equivalent conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19066.7 Categorical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19066.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19066.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

67 Shrinking space 19267.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

68 Simply connected at innity 19368.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

69 Simply connected space 19469.1 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19469.2 Formal denition and equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19569.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19569.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19769.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19769.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

70 Sub-Stonean space 19870.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19870.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19870.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

71 Subspace topology 19971.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19971.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

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71.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20071.4 Preservation of topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20171.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20171.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

72 Supercompact space 20272.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20272.2 Some Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20272.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

73 T1 space 20473.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20473.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20473.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20573.4 Generalisations to other kinds of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20673.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

74 Tenseaspectmood 20774.1 Creoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

74.1.1 Hawaiian Creole English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20774.2 Modern Greek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20874.3 Slavic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

74.3.1 Russian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20974.4 Romance languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

74.4.1 French . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20974.4.2 Italian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20974.4.3 Portuguese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21074.4.4 Spanish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

74.5 Germanic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21174.5.1 German . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21174.5.2 Danish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21174.5.3 Dutch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21174.5.4 Icelandic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21274.5.5 English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

74.6 Basque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21674.7 Hawaiian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21674.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21674.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

75 Topological manifold 21875.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21875.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21875.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

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75.3.1 The Hausdor axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21975.3.2 Compactness and countability axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21975.3.3 Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

75.4 Coordinate charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22075.5 Classication of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22075.6 Manifolds with boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22175.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22175.8 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22175.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

76 Topological property 22276.1 Common topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

76.1.1 Cardinal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22276.1.2 Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22276.1.3 Countability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22376.1.4 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22376.1.5 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22476.1.6 Metrizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22476.1.7 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

76.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22576.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22576.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

77 Topological space 22677.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

77.1.1 Neighbourhoods denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22677.1.2 Open sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22777.1.3 Closed sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22877.1.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

77.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22877.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22877.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22977.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23077.6 Classication of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23077.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23077.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23077.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23077.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23177.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23177.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23177.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

xvi CONTENTS

78 Toronto space 23378.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

79 Totally disconnected space 23479.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23479.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23479.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23579.4 Constructing a disconnected space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23579.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23579.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

80 Ultraconnected space 23680.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23680.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23680.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

81 Uniformizable space 23781.1 Induced uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23781.2 Fine uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23781.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

82 Uses of English verb forms 23982.1 Inected forms of verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23982.2 Verbs in combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23982.3 Tenses, aspects and moods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

82.3.1 Tenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24082.3.2 Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24182.3.3 Moods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

82.4 Active and passive voice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24382.5 Negation and questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24482.6 Modal verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24482.7 Uses of verb combination types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

82.7.1 Simple present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24482.7.2 Present progressive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24682.7.3 Present perfect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24682.7.4 Present perfect progressive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24782.7.5 Simple past . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24882.7.6 Past progressive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24882.7.7 Past perfect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24982.7.8 Past perfect progressive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25082.7.9 Simple future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25082.7.10 Future progressive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25182.7.11 Future perfect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

CONTENTS xvii

82.7.12 Future perfect progressive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25182.7.13 Simple conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25282.7.14 Conditional progressive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25282.7.15 Conditional perfect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25382.7.16 Conditional perfect progressive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

82.8 Have got and can see . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25382.9 Been and gone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25482.10Conditional sentences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25482.11Expressions of wish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25582.12Indirect speech . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25682.13Dependent clauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25782.14Uses of nonnite verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

82.14.1 Bare innitive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25782.14.2 To-innitive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25882.14.3 Present participle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26082.14.4 Past participle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26182.14.5 Gerund . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26282.14.6 Perfect and progressive nonnite constructions . . . . . . . . . . . . . . . . . . . . . . . . 263

82.15Deverbal uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26382.16Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26482.17References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

83 Volterra space 26683.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

84 Weak Hausdor space 26784.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

85 Zero-dimensional space 26885.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26885.2 Properties of spaces with covering dimension zero . . . . . . . . . . . . . . . . . . . . . . . . . . 26885.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26885.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

86 -compact space 27086.1 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27086.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27086.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27186.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

87 -bounded space 27287.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27287.2 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 273

xviii CONTENTS

87.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27387.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27987.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

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

Alexandrov topology

In topology, an Alexandrov space (or Alexandrov-discrete space) is a topological space in which the intersectionof any family of open sets is open. It is an axiom of topology that the intersection of any nite family of open sets isopen. In an Alexandrov space the nite restriction is dropped.Alexandrov topologies are uniquely determined by their specialization preorders. Indeed, given any preorder on aset X, there is a unique Alexandrov topology on X for which the specialization preorder is . The open sets are justthe upper sets with respect to . Thus, Alexandrov topologies on X are in one-to-one correspondence with preorderson X.Alexandrov spaces are also called nitely generated spaces since their topology is uniquely determined by the familyof all nite subspaces. Alexandrov spaces can be viewed as a generalization of nite topological spaces.

2.1 Characterizations of Alexandrov topologiesAlexandrov topologies have numerous characterizations. Let X = be a topological space. Then the followingare equivalent:

Open and closed set characterizations: Open set. An arbitrary intersection of open sets in X is open. Closed set. An arbitrary union of closed sets in X is closed.

Neighbourhood characterizations: Smallest neighbourhood. Every point of X has a smallest neighbourhood. Neighbourhood lter. The neighbourhood lter of every point in X is closed under arbitrary intersec-tions.

Interior and closure algebraic characterizations: Interior operator. The interior operator of X distributes over arbitrary intersections of subsets. Closure operator. The closure operator of X distributes over arbitrary unions of subsets.

Preorder characterizations: Specialization preorder. T is the nest topology consistent with the specialization preorder of X i.e.the nest topology giving the preorder satisfying x y if and only if x is in the closure of {y} in X.

Open up-set. There is a preorder such that the open sets of X are precisely those that are upwardlyclosed i.e. if x is in the set and x y then y is in the set. (This preorder will be precisely the specializationpreorder.)

2

2.2. DUALITY WITH PREORDERED SETS 3

Closed down-set. There is a preorder such that the closed sets of X are precisely those that aredownwardly closed i.e. if x is in the set and y x then y is in the set. (This preorder will be precisely thespecialization preorder.)

Upward interior. A point x lies in the interior of a subset S of X if and only if there is a point y in Ssuch that y x where is the specialization preorder i.e. y lies in the closure of {x}.

Downward closure. A point x lies in the closure of a subset S of X if and only if there is a point y in Ssuch that x y where is the specialization preorder i.e. x lies in the closure of {y}.

Finite generation and category theoretic characterizations: Finite closure. A point x lies within the closure of a subset S of X if and only if there is a nite subset

F of S such that x lies in the closure of F. Finite subspace. T is coherent with the nite subspaces of X. Finite inclusion map. The inclusion maps fi : Xi X of the nite subspaces of X form a nal sink. Finite generation. X is nitely generated i.e. it is in the nal hull of the nite spaces. (This means thatthere is a nal sink fi : Xi X where each Xi is a nite topological space.)

Topological spaces satisfying the above equivalent characterizations are called nitely generated spaces or Alexan-drov spaces and their topology T is called the Alexandrov topology, named after the Russian mathematician PavelAlexandrov who rst investigated them.

2.2 Duality with preordered sets

2.2.1 The Alexandrov topology on a preordered setGiven a preordered set X = hX;i we can dene an Alexandrov topology on X by choosing the open sets to bethe upper sets:

= fG X : 8x; y 2 X x 2 G ^ x y ! y 2 G; gWe thus obtain a topological space T(X) = hX; i .The corresponding closed sets are the lower sets:

fS X : 8x; y 2 X x 2 S ^ y x ! y 2 S; g

2.2.2 The specialization preorder on a topological spaceGiven a topological space X = the specialization preorder on X is dened by:

xy if and only if x is in the closure of {y}.

We thus obtain a preordered set W(X) = .

2.2.3 Equivalence between preorders and Alexandrov topologiesFor every preordered set X = we always have W(T(X)) = X, i.e. the preorder of X is recovered from thetopological space T(X) as the specialization preorder. Moreover for every Alexandrov space X, we have T(W(X)) =X, i.e. the Alexandrov topology of X is recovered as the topology induced by the specialization preorder.However for a topological space in general we do not have T(W(X)) = X. Rather T(W(X)) will be the set X with aner topology than that of X (i.e. it will have more open sets).

4 CHAPTER 2. ALEXANDROV TOPOLOGY

2.2.4 Equivalence between monotony and continuityGiven a monotone function

f : XY

between two preordered sets (i.e. a function

f : XY

between the underlying sets such that xy in X implies f(x)f(y) in Y), let

T(f) : T(X)T(Y)

be the same map as f considered as a map between the corresponding Alexandrov spaces. Then

T(f) : T(X)T(Y)

is a continuous map.Conversely given a continuous map

f : XY

between two topological spaces, let

W(f) : W(X)W(Y)

be the same map as f considered as a map between the corresponding preordered sets. Then

W(f) : W(X)W(Y)

is a monotone function.Thus a map between two preordered sets is monotone if and only if it is a continuous map between the correspondingAlexandrov spaces. Conversely a map between two Alexandrov spaces is continuous if and only if it is a monotonefunction between the corresponding preordered sets.Notice however that in the case of topologies other than the Alexandrov topology, we can have a map between twotopological spaces that is not continuous but which is nevertheless still a monotone function between the correspondingpreordered sets. (To see this consider a non-Alexandrov space X and consider the identity map

i : XT(W(X)).)

2.2.5 Category theoretic description of the dualityLet Set denote the category of sets and maps. Let Top denote the category of topological spaces and continuousmaps; and let Pro denote the category of preordered sets and monotone functions. Then

T : ProTop and

W : TopPro

are concrete functors over Set which are left and right adjoints respectively.Let Alx denote the full subcategory of Top consisting of the Alexandrov spaces. Then the restrictions

2.3. HISTORY 5

T : ProAlx and

W : AlxPro

are inverse concrete isomorphisms over Set.Alx is in fact a bico-reective subcategory of Top with bico-reector TW : TopAlx. This means that given atopological space X, the identity map

i : T(W(X))X

is continuous and for every continuous map

f : YX

where Y is an Alexandrov space, the composition

i 1f : YT(W(X))

is continuous.

2.2.6 Relationship to the construction of modal algebras from modal framesGiven a preordered set X, the interior operator and closure operator of T(X) are given by:

Int(S) = { x X : for all y X, xy implies y S }, for all S X

Cl(S) = { x X : there exists a y S with xy } for all S X

Considering the interior operator and closure operator to be modal operators on the power set Boolean algebra of X,this construction is a special case of the construction of a modal algebra from a modal frame i.e. a set with a singlebinary relation. (The latter construction is itself a special case of a more general construction of a complex algebrafrom a relational structure i.e. a set with relations dened on it.) The class of modal algebras that we obtain in thecase of a preordered set is the class of interior algebrasthe algebraic abstractions of topological spaces.

2.3 HistoryAlexandrov spaces were rst introduced in 1937 by P. S. Alexandrov under the name discrete spaces, where heprovided the characterizations in terms of sets and neighbourhoods.[1] The name discrete spaces later came to be usedfor topological spaces in which every subset is open and the original concept lay forgotten. With the advancement ofcategorical topology in the 1980s, Alexandrov spaces were rediscovered when the concept of nite generation wasapplied to general topology and the name nitely generated spaces was adopted for them. Alexandrov spaces werealso rediscovered around the same time in the context of topologies resulting from denotational semantics and domaintheory in computer science.In 1966 Michael C. McCord and A. K. Steiner each independently observed a duality between partially ordered setsand spaces which were precisely the T0 versions of the spaces that Alexandrov had introduced.[2][3] P. Johnstonereferred to such topologies as Alexandrov topologies.[4] F. G. Arenas independently proposed this name for thegeneral version of these topologies.[5] McCord also showed that these spaces are weak homotopy equivalent to theorder complex of the corresponding partially ordered set. Steiner demonstrated that the duality is a contravariantlattice isomorphism preserving arbitrary meets and joins as well as complementation.It was also a well known result in the eld of modal logic that a duality exists between nite topological spaces andpreorders on nite sets (the nite modal frames for the modal logic S4). C. Naturman extended these results to aduality between Alexandrov spaces and preorders in general, providing the preorder characterizations as well as theinterior and closure algebraic characterizations.[6]

A systematic investigation of these spaces from the point of view of general topology which had been neglected sincethe original paper by Alexandrov, was taken up by F.G. Arenas.[5]

6 CHAPTER 2. ALEXANDROV TOPOLOGY

2.4 See also P-space, a space satisfying the weaker condition that countable intersections of open sets are open

2.5 References[1] Alexandro, P. (1937). Diskrete Rume. Mat. Sb. (N.S.) (in German) 2: 501518.

[2] McCord, M. C. (1966). Singular homology and homotopy groups of nite topological spaces. Duke Mathematical Journal33 (3): 465474. doi:10.1215/S0012-7094-66-03352-7.

[3] Steiner, A. K. (1966). The Lattice of Topologies: Structure and Complementation. Transactions of the American Math-ematical Society 122 (2): 379398. doi:10.2307/1994555. ISSN 0002-9947.

[4] Johnstone, P. T. (1986). Stone spaces (1st paperback ed.). New York: Cambridge University Press. ISBN 0-521-33779-8.

[5] Arenas, F. G. (1999). Alexandro spaces (PDF). Acta Math. Univ. Comenianae 68 (1): 1725.

[6] Naturman, C. A. (1991). Interior Algebras and Topology. Ph.D. thesis, University of Cape Town Department of Mathe-matics.

Chapter 3

Annulus (mathematics)

O

r

d

R

A B

An annulus

In mathematics, an annulus (the Latin word for little ring, with plural annuli) is a ring-shaped object, especially aregion bounded by two concentric circles. The adjectival form is annular (as in annular eclipse).The open annulus is topologically equivalent to both the open cylinder S1 (0,1) and the punctured plane.

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8 CHAPTER 3. ANNULUS (MATHEMATICS)

The area of an annulus is the dierence in the areas of the larger circle of radius R and the smaller one of radius r:

A = R2 r2 = (R2 r2) :

The area of an annulus can be obtained from the length of the longest interval that can lie completely inside theannulus, 2*d in the accompanying diagram. This can be proven by the Pythagorean theorem; the length of thelongest interval that can lie completely inside the annulus will be tangent to the smaller circle and form a right anglewith its radius at that point. Therefore d and r are the sides of a right angled triangle with hypotenuse R and the areais given by:

A = R2 r2 = d2 :

The area can also be obtained via calculus by dividing the annulus up into an innite number of annuli of innitesimalwidth d and area 2 d and then integrating from = r to = R:

A =

Z Rr

2 d = R2 r2 :

The area of an annulus sector of angle , with measured in radians, is given by:

A =

2

R2 r2

3.1 Complex structureIn complex analysis an annulus ann(a; r, R) in the complex plane is an open region dened by:

r < jz aj < R:

If r is 0, the region is known as the punctured disk of radius R around the point a.As a subset of the complex plane, an annulus can be considered as a Riemann surface. The complex structure of anannulus depends only on the ratio r/R. Each annulus ann(a; r, R) can be holomorphically mapped to a standard onecentered at the origin and with outer radius 1 by the map

z 7! z aR

:

The inner radius is then r/R < 1.The Hadamard three-circle theorem is a statement about the maximum value a holomorphic function may take insidean annulus.

3.2 See also Annulus theorem (or conjecture) Spherical shell Torus List of geometric shapes

3.3. EXTERNAL LINKS 9

3.3 External links Annulus denition and properties With interactive animation Area of an annulus, formula With interactive animation

Chapter 4

Baire space

For the concept in set theory, see Baire space (set theory).

In mathematics, a Baire space is a topological space that has enough points that every intersection of a countablecollection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdor spacesare examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of Ren-LouisBaire who introduced the concept.

4.1 MotivationIn an arbitrary topological space, the class of closed sets with empty interior consists precisely of the boundaries ofdense open sets. These sets are, in a certain sense, negligible. Some examples are nite sets in , smooth curves inthe plane, and proper ane subspaces in a Euclidean space. If a topological space is a Baire space then it is large,meaning that it is not a countable union of negligible subsets. For example, the three-dimensional Euclidean space isnot a countable union of its ane planes.

4.2 DenitionThe precise denition of a Baire space has undergone slight changes throughout history, mostly due to prevailingneeds and viewpoints. First, we give the usual modern denition, and then we give a historical denition which iscloser to the denition originally given by Baire.

4.2.1 Modern denitionA Baire space is a topological space in which the union of every countable collection of closed sets with emptyinterior has empty interior.This denition is equivalent to each of the following conditions:

Every intersection of countably many dense open sets is dense. The interior of every union of countably many closed nowhere dense sets is empty. Whenever the union of countably many closed subsets of X has an interior point, then one of the closed subsetsmust have an interior point.

4.2.2 Historical denitionMain article: Meagre set

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4.3. EXAMPLES 11

In his original denition, Baire dened a notion of category (unrelated to category theory) as follows.A subset of a topological space X is called

nowhere dense in X if the interior of its closure is empty of rst category or meagre in X if it is a union of countably many nowhere dense subsets of second category or nonmeagre in X if it is not of rst category in X

The denition for a Baire space can then be stated as follows: a topological spaceX is a Baire space if every non-emptyopen set is of second category in X. This denition is equivalent to the modern denition.A subset A of X is comeagre if its complementX nA is meagre. A topological space X is a Baire space if and onlyif every comeager subset of X is dense.

4.3 Examples The space R of real numbers with the usual topology, is a Baire space, and so is of second category in itself.The rational numbers are of rst category and the irrational numbers are of second category in R .

The Cantor set is a Baire space, and so is of second category in itself, but it is of rst category in the interval[0; 1] with the usual topology.

Here is an example of a set of second category in R with Lebesgue measure 0.

1\m=1

1[n=1

rn 1

2n+m; rn +

1

2n+m

where frng1n=1 is a sequence that enumerates the rational numbers.

Note that the space of rational numbers with the usual topology inherited from the reals is not a Baire space,since it is the union of countably many closed sets without interior, the singletons.

4.4 Baire category theoremMain article: Baire category theorem

The Baire category theorem gives sucient conditions for a topological space to be a Baire space. It is an importanttool in topology and functional analysis.

(BCT1) Every complete metric space is a Baire space. More generally, every topological space which ishomeomorphic to an open subset of a complete pseudometric space is a Baire space. In particular, everycompletely metrizable space is a Baire space.

(BCT2) Every locally compact Hausdor space (or more generally every locally compact sober space) is aBaire space.

BCT1 shows that each of the following is a Baire space:

The space R of real numbers The space of irrational numbers, which is homeomorphic to the Baire space of set theory The Cantor set Indeed, every Polish space

BCT2 shows that every manifold is a Baire space, even if it is not paracompact, and hence not metrizable. Forexample, the long line is of second category.

12 CHAPTER 4. BAIRE SPACE

4.5 Properties Every non-empty Baire space is of second category in itself, and every intersection of countably many denseopen subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topologicaldisjoint sum of the rationals and the unit interval [0, 1].

Every open subspace of a Baire space is a Baire space.

Given a family of continuous functions fn:XY with pointwise limit f:XY. If X is a Baire space then thepoints where f is not continuous is a meagre set in X and the set of points where f is continuous is dense in X.A special case of this is the uniform boundedness principle.

A closed subset of a Baire space is not necessarily Baire.

The product of two Baire spaces is not necessarily Baire. However, there exist sucient conditions that willguarantee that a product of arbitrarily many Baire spaces is again Baire.

4.6 See also BanachMazur game Descriptive set theory Baire space (set theory)

4.7 References

4.8 Sources Munkres, James, Topology, 2nd edition, Prentice Hall, 2000. Baire, Ren-Louis (1899), Sur les fonctions de variables relles, Annali di Mat. Ser. 3 3, 1123.

4.9 External links Encyclopaedia of Mathematics article on Baire space Encyclopaedia of Mathematics article on Baire theorem

Chapter 5

Collectionwise Hausdor space

In mathematics, in the eld of topology, a topological space is said to be collectionwise Hausdor if given any closeddiscrete collection of points in the topological space, there are pairwise disjoint open sets containing the points.[1] Aclosed discrete set S of a topology X is one where every point of X has a neighborhood that intersects at most onepoint from S. Every T1 space which is collectionwise Hausdor is also Hausdor.Metrizable spaces are collectionwise normal spaces and are hence, in particular, collectionwise Hausdor.

5.1 References[1] FD Tall, The density topology, Pacic Journal of Mathematics, 1976

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Chapter 6

Collectionwise normal space

In mathematics, a topological spaceX is called collectionwise normal if for every discrete family Fi (i I) of closedsubsets of X there exists a pairwise disjoint family of open sets Ui (i I), such that Fi Ui. A family F of subsetsof X is called discrete when every point of X has a neighbourhood that intersects at most one of the sets from F .An equivalent denition demands that the above Ui (i I) are themselves a discrete family, which is stronger thanpairwise disjoint.Many authors assume that X is also a T1 space as part of the denition, i. e., for every pair of distinct points, eachhas an open neighborhood not containing the other. A collectionwise normal T1 space is a collectionwise Hausdorspace.Every collectionwise normal space is normal (i. e., any two disjoint closed sets can be separated by neighbourhoods),and every paracompact space (i. e., every topological space in which every open cover admits a locally nite openrenement) is collectionwise normal. The property is therefore intermediate in strength between paracompactnessand normality.Every metrizable space (i. e., every topological space that is homeomorphic to a metric space) is collectionwisenormal. The Moore metrisation theorem states that every collectionwise normal Moore space is metrizable.An F-set in a collectionwise normal space is also collectionwise normal in the subspace topology. In particular, thisholds for closed subsets.

6.1 References Engelking, Ryszard, General Topology, Heldermann Verlag Berlin, 1989. ISBN 3-88538-006-4

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Chapter 7

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

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16 CHAPTER 7. COMPACT SPACE

ArzelAscoli theorem or the Peano existence theorem, in which one is able to conclude the existence of a functionwith some required properties as a limiting case of some more elementary construction.Various equivalent notions of compactness, including sequential compactness and limit point compactness, can bedeveloped in general metric spaces. In general topological spaces, however, dierent notions of compactness are notnecessarily equivalent. The most useful notion, which is the standard denition of the unqualied term compactness,is phrased in terms of the existence of nite families of open sets that "cover" the space in the sense that each pointof the space must lie in some set contained in the family. This