Dense set

Dense setFrom Wikipedia, the free encyclopedia

Contents

1 Adherent point 11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Alexandro extension 22.1 Example: inverse stereographic projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 The Alexandro extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 The one-point compactication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Appert topology 53.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Related topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Axiom of countability 74.1 Important examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 Relationships with each other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.3 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Baire category theorem 95.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.2 Relation to the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.3 Uses of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.4 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

i

ii CONTENTS

5.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 Baire function 126.1 Classication of Baire functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.2 Baire class 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.3 Baire class 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.4 Baire class 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7 Baire set 147.1 Basic denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

7.1.1 First denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.1.2 Second denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.1.3 Third denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.2.1 The dierent denitions of Baire sets are not equivalent . . . . . . . . . . . . . . . . . . . 157.2.2 A Borel set that is not a Baire set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8 Baire space 178.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8.2.1 Modern denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.2.2 Historical denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.4 Baire category theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.8 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

9 BanachMazur game 209.1 Denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.2 A simple proof: winning strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

10 Base (topology) 22

CONTENTS iii

10.1 Simple properties of bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2210.2 Objects dened in terms of bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.3 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.4 Base for the closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.5 Weight and character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

10.5.1 Increasing chains of open sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

11 Boundary (topology) 2611.1 Common denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.4 Boundary of a boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

12 Cardinality 3012.1 Comparing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

12.1.1 Denition 1: | A | = | B | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.1.2 Denition 2: | A | | B | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.1.3 Denition 3: | A | < | B | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

12.2 Cardinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.3 Finite, countable and uncountable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.4 Innite sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

12.4.1 Cardinality of the continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.5 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.6 Union and intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

13 Closed set 3513.1 Equivalent denitions of a closed set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.2 Properties of closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.3 Examples of closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.4 More about closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

14 Closure (topology) 3714.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

14.1.1 Point of closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

iv CONTENTS

14.1.2 Limit point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3714.1.3 Closure of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

14.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3814.3 Closure operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3914.4 Facts about closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3914.5 Categorical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

15 Compact space 4115.1 Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4215.2 Basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4315.3 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

15.3.1 Open cover denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4315.3.2 Equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.3.3 Compactness of subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

15.4 Properties of compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4515.4.1 Functions and compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4515.4.2 Compact spaces and set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4515.4.3 Ordered compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

15.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4615.5.1 Algebraic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

15.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4715.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4815.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4815.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

16 Compactication (mathematics) 5016.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5016.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

16.2.1 Alexandro one-point compactication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5116.2.2 Stoneech compactication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

16.3 Projective space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5116.4 Compactication and discrete subgroups of Lie groups . . . . . . . . . . . . . . . . . . . . . . . . 5216.5 Other compactication theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5216.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

17 Complete metric space 5317.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5317.2 Some theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

CONTENTS v

17.3 Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5417.4 Topologically complete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5517.5 Alternatives and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5517.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5517.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5617.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

18 Countable set 5718.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.4 Formal denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5818.5 Minimal model of set theory is countable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.6 Total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

19 Countably compact space 6519.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

20 Dense set 6620.1 Density in metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6620.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6620.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.4 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

20.6.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6820.6.2 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

21 Diophantine approximation 6921.1 Best Diophantine approximations of a real number . . . . . . . . . . . . . . . . . . . . . . . . . . 6921.2 Measure of the accuracy of approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

21.2.1 Badly approximable numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7021.3 Lower bounds for Diophantine approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

21.3.1 Approximation of a rational by other rationals . . . . . . . . . . . . . . . . . . . . . . . . 7021.3.2 Approximation of algebraic numbers, Liouvilles result . . . . . . . . . . . . . . . . . . . 7121.3.3 Approximation of algebraic numbers, ThueSiegelRoth theorem . . . . . . . . . . . . . 71

vi CONTENTS

21.3.4 Simultaneous approximations of algebraic numbers . . . . . . . . . . . . . . . . . . . . . 7121.3.5 Eective bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

21.4 Upper bounds for Diophantine approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7221.4.1 General upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7221.4.2 Equivalent real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7221.4.3 Lagrange spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

21.5 Khinchins theorem and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7321.6 Uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7421.7 Unsolved problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7421.8 Recent developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7421.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7421.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7421.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7521.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

22 Embedding 7722.1 Topology and geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

22.1.1 General topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7722.1.2 Dierential topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7722.1.3 Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

22.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7822.2.1 Field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7822.2.2 Universal algebra and model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

22.3 Order theory and domain theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7922.4 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

22.4.1 Normed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7922.5 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7922.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8022.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8022.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8022.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

23 Fort space 8223.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8223.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

24 General topology 8324.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8424.2 A topology on a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

24.2.1 Basis for a topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8424.2.2 Subspace and quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8424.2.3 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

CONTENTS vii

24.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8524.3.1 Alternative denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8624.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8724.3.3 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8724.3.4 Dening topologies via continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . 88

24.4 Compact sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8824.5 Connected sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

24.5.1 Connected components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8924.5.2 Disconnected spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8924.5.3 Path-connected sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

24.6 Products of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9024.7 Separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9124.8 Countability axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9224.9 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9224.10Baire category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9324.11Main areas of research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

24.11.1 Continuum theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9324.11.2 Pointless topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9324.11.3 Dimension theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9324.11.4 Topological algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9324.11.5 Metrizability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9424.11.6 Set-theoretic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

24.12See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9424.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9424.14Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

25 Glossary of topology 9625.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9725.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9825.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9825.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10025.5 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10025.6 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10025.7 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10125.8 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10125.9 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10225.10K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10225.11L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10325.12M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10325.13N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10425.14O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10525.15P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

viii CONTENTS

25.16Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10625.17R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10725.18S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10725.19T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10825.20U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10925.21W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11025.22Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11025.23References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11025.24External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

26 Hausdor space 11226.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11226.2 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11326.3 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11326.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11326.5 Preregularity versus regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11426.6 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11426.7 Algebra of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11526.8 Academic humour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11526.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11526.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11526.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

27 Interior (topology) 11627.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

27.1.1 Interior point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11727.1.2 Interior of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

27.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11727.3 Interior operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11827.4 Exterior of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11827.5 Interior-disjoint shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11927.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11927.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11927.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

28 Intersection 12128.1 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12228.2 Examples in classical geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12228.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12228.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12228.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

29 List of general topology topics 123

CONTENTS ix

29.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12329.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12329.3 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

29.3.1 Compactness and countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12429.3.2 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12429.3.3 Separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

29.4 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12529.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12529.6 Uniform spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12629.7 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12629.8 Topology and order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12729.9 Descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12729.10Dimension theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12729.11Combinatorial topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12729.12Foundations of algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12829.13Topology and algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12829.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

30 Locally compact space 12930.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12930.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

30.2.1 Compact Hausdor spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13030.2.2 Locally compact Hausdor spaces that are not compact . . . . . . . . . . . . . . . . . . . 13030.2.3 Hausdor spaces that are not locally compact . . . . . . . . . . . . . . . . . . . . . . . . 13030.2.4 Non-Hausdor examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

30.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13130.3.1 The point at innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13130.3.2 Locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

30.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13230.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

31 Natural number 13331.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

31.1.1 Modern denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13531.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13531.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

31.3.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13531.3.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13531.3.3 Relationship between addition and multiplication . . . . . . . . . . . . . . . . . . . . . . . 13631.3.4 Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13631.3.5 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13631.3.6 Algebraic properties satised by the natural numbers . . . . . . . . . . . . . . . . . . . . . 136

x CONTENTS

31.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13731.5 Formal denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

31.5.1 Peano axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13731.5.2 Constructions based on set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

31.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13931.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13931.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14031.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

32 Neighbourhood (mathematics) 14432.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14532.2 In a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14532.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14732.4 Topology from neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14732.5 Uniform neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14732.6 Deleted neighbourhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14732.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14732.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

33 Normal space 14933.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14933.2 Examples of normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15033.3 Examples of non-normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15033.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15133.5 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15133.6 Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15133.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

34 Open set 15234.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15334.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

34.2.1 Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15434.2.2 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15434.2.3 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

34.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15434.4 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15434.5 Notes and cautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

34.5.1 Open is dened relative to a particular topology . . . . . . . . . . . . . . . . . . . . . . 15534.5.2 Open and closed are not mutually exclusive . . . . . . . . . . . . . . . . . . . . . . . . . 155

34.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15534.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15534.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

CONTENTS xi

35 Pavel Alexandrov 15735.1 Honours and awards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15735.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15835.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

36 Rational number 15936.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15936.2 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

36.2.1 Embedding of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16036.2.2 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16036.2.3 Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16036.2.4 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16036.2.5 Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16036.2.6 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16036.2.7 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16036.2.8 Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16136.2.9 Exponentiation to integer power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

36.3 Continued fraction representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16136.4 Formal construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16136.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16336.6 Real numbers and topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16436.7 p-adic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16436.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16436.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16436.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

37 Real number 16637.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16737.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

37.2.1 Axiomatic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16837.2.2 Construction from the rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

37.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16937.3.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16937.3.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16937.3.3 The complete ordered eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17037.3.4 Advanced properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

37.4 Applications and connections to other areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17137.4.1 Real numbers and logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17137.4.2 In physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17237.4.3 In computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17237.4.4 Reals in set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

37.5 Vocabulary and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

xii CONTENTS

37.6 Generalizations and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17337.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17337.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17337.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17437.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

38 Ren-Louis Baire 17538.1 Schooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17538.2 Illness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17538.3 Contributions to Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17538.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17638.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17638.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

39 Stoneech compactication 17739.1 Universal property and functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17739.2 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

39.2.1 Construction using products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17839.2.2 Construction using the unit interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17839.2.3 Construction using ultralters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17839.2.4 Construction using C*-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

39.3 The Stoneech compactication of the natural numbers . . . . . . . . . . . . . . . . . . . . . . . 17939.3.1 An application: the dual space of the space of bounded sequences of reals . . . . . . . . . 17939.3.2 Addition on the Stoneech compactication of the naturals . . . . . . . . . . . . . . . . . 180

39.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18039.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18139.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18139.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

40 Subset 18240.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18340.2 and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18340.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18340.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18440.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18440.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18440.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

41 Topological space 18641.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

41.1.1 Neighbourhoods denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18641.1.2 Open sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18741.1.3 Closed sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

CONTENTS xiii

41.1.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18841.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18841.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18841.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18941.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19041.6 Classication of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19041.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19041.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19041.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19041.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19141.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19141.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19141.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

42 Union (set theory) 19342.1 Union of two sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19342.2 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19442.3 Finite unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19542.4 Arbitrary unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

42.4.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19542.4.2 Union and intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

42.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19642.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19642.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19642.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 197

42.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19742.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20342.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Chapter 1

Adherent point

In mathematics, an adherent point (also closure point or point of closure or contact point)[1] of a subset A of atopological space X, is a point x in X such that every open set containing x contains at least one point of A . A pointx is an adherent point for A if and only if x is in the closure of A.This denition diers from that of a limit point, in that for a limit point it is required that every open set containingx contains at least one point of A dierent from x. Thus every limit point is an adherent point, but the converse is nottrue. An adherent point of A is either a limit point of A or an element of A (or both). An adherent point which is nota limit point is an isolated point.Intuitively, having an open set A dened as the area within (but not including) some boundary, the adherent points ofA are those of A including the boundary.

1.1 Examples If S is a subset of R which is bounded above, then sup S is adherent to S. A subset S of a metric space M contains all of its adherent points if, and only if, S is closed in M. In the interval (a, b], a is an adherent point that is not in the interval, with usual topology of R.

1.2 Notes[1] Steen, p. 5; Lipschutz, p. 69; Adamson, p. 15.

1.3 References Adamson, Iain T., A General Topology Workbook, Birkhuser Boston; 1st edition (November 29, 1995). ISBN978-0-8176-3844-3.

Apostol, Tom M., Mathematical Analysis, Addison Wesley Longman; second edition (1974). ISBN 0-201-00288-4

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

L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, (1970) Holt, Rinehart and Winston, Inc.. This article incorporates material from Adherent point on PlanetMath, which is licensed under the Creative

Commons Attribution/Share-Alike License.

1

Chapter 2

Alexandro extension

In mathematical eld of topology, the Alexandro extension is a way to extend a noncompact topological space byadjoining a single point in such a way that the resulting space is compact. It is named for the Russian mathematicianPavel Alexandrov.More precisely, let X be a topological space. Then the Alexandro extension of X is a certain compact space X*together with an open embedding c : X X* such that the complement of X in X* consists of a single point, typicallydenoted . The map c is a Hausdor compactication if and only if X is a locally compact, noncompact Hausdorspace. For such spaces the Alexandro extension is called the one-point compactication or Alexandro com-pactication. The advantages of the Alexandro compactication lie in its simple, often geometrically meaningfulstructure and the fact that it is in a precise sense minimal among all compactications; the disadvantage lies in thefact that it only gives a Hausdor compactication on the class of locally compact, noncompact Hausdor spaces,unlike the Stoneech compactication which exists for any Tychono space, a much larger class of spaces.

2.1 Example: inverse stereographic projection

A geometrically appealing example of one-point compactication is given by the inverse stereographic projection.Recall that the stereographic projection S gives an explicit homeomorphism from the unit sphere minus the north pole(0,0,1) to the Euclidean plane. The inverse stereographic projection S1 : R2 ,! S2 is an open, dense embeddinginto a compact Hausdor space obtained by adjoining the additional point1 = (0; 0; 1) . Under the stereographicprojection latitudinal circles z = c get mapped to planar circles r =

p(1 + c)/(1 c) . It follows that the deleted

neighborhood basis of (1; 0; 0) given by the punctured spherical caps c z < 1 corresponds to the complementsof closed planar disks r p(1 + c)/(1 c) . More qualitatively, a neighborhood basis at1 is furnished by thesets S1(R2 nK) [ f1g as K ranges through the compact subsets of R2 . This example already contains the keyconcepts of the general case.

2.2 Motivation

Let c : X ,! Y be an embedding from a topological space X to a compact Hausdor topological space Y, withdense image and one-point remainder f1g = Y n c(X) . Then c(X) is open in a compact Hausdor space so islocally compact Hausdor, hence its homeomorphic preimage X is also locally compact Hausdor. Moreover, ifX were compact then c(X) would be closed in Y and hence not dense. Thus a space can only admit a one-pointcompactication if it is locally compact, noncompact and Hausdor. Moreover, in such a one point compacticationthe image of a neighborhood basis for x in X gives a neighborhood basis for c(x) in c(X), andbecause a subset ofa compact Hausdor space is compact if and only if it is closedthe open neighborhoods of 1 must be all setsobtained by adjoining1 to the image under c of a subset of X with compact complement.

2

2.3. THE ALEXANDROFF EXTENSION 3

2.3 The Alexandro extensionLet X be any topological space, and let1 be any object which is not already an element of X. Put X = X [ f1g, and topologize X by taking as open sets all the open subsets U of X together with all subsets V which contain1and such that X n V is closed and compact, (Kelley 1975, p. 150).The inclusion map c : X ! X is called the Alexandro extension of X (Willard, 19A).The above properties all follow from the above discussion:

The map c is continuous and open: it embeds X as an open subset of X .

The space X is compact.

The image c(X) is dense in X , if X is noncompact.

The space X is Hausdor if and only if X is Hausdor and locally compact.

2.4 The one-point compacticationIn particular, the Alexandro extension c : X ! X is a compactication of X if and only if X is Hausdor,noncompact and locally compact. In this case it is called the one-point compactication or Alexandro compact-ication of X. Recall from the above discussion that any compactication with one point remainder is necessarily(isomorphic to) the Alexandro compactication.Let X be any noncompact Tychono space. Under the natural partial ordering on the set C(X) of equivalence classesof compactications, any minimal element is equivalent to the Alexandro extension (Engelking, Theorem 3.5.12).It follows that a noncompact Tychono space admits a minimal compactication if and only if it is locally compact.

2.5 Further examples The one-point compactication of the set of positive integers is homeomorphic to the space consisting of K ={0} U {1/n | n is a positive integer.} with the order topology.

The one-point compactication of n-dimensional Euclidean space Rn is homeomorphic to the n-sphere Sn. Asabove, the map can be given explicitly as an n-dimensional inverse stereographic projection.

Since the closure of a connected subset is connected, the Alexandro extension of a noncompact connectedspace is connected. However a one-point compactication may connect a disconnected space: for instancethe one-point compactication of the disjoint union of copies of the interval (0,1) is a wedge of circles.

The Alexandro extension can be viewed as a functor from the category of topological spaces to the categorywhose objects are continuous maps c : X ! Y and for which the morphisms from c1 : X1 ! Y1 toc2 : X2 ! Y2 are pairs of continuous maps fX : X1 ! X2; fY : Y1 ! Y2 such that fY c1 = c2 fX . Inparticular, homeomorphic spaces have isomorphic Alexandro extensions.

A sequence fang in a topological space X converges to a point a in X , if and only if the map f : N ! Xgiven by f(n) = an for n in N and f(1) = a is continuous. Here N has the discrete topology.

Polyadic spaces are dened as topological spaces that are the continuous image of the power of a one-pointcompactication of a discrete, locally compact Haussdor space.

Space of continuous functions C () on a locally compact Hausdro space is locally compact but can bemade compact if and only if we include the single point f(x) = 1 for all x

4 CHAPTER 2. ALEXANDROFF EXTENSION

2.6 See also Wallman compactication End (topology) Riemann sphere Normal space Stereographic projection Pointed set

2.7 References P.S. Alexandro (1924), "ber die Metrisation der im Kleinen kompakten topologischen Rume,Math. Ann.92 (3-4): 294301, doi:10.1007/BF01448011, JFM 50.0128.04

Ronald Brown (1973), Sequentially proper maps and a sequential compactication, J. London Math Soc. (2)7: 515522, doi:10.1112/jlms/s2-7.3.515, Zbl 0269.54015

Engelking, Ryszard (1989), General Topology, Helderman Verlag Berlin, ISBN 978-0-201-08707-9, MR1039321

Fedorchuk, V.V. (2001), Aleksandrov compactication, in Hazewinkel, Michiel, Encyclopedia of Mathe-matics, Springer, ISBN 978-1-55608-010-4

Kelley, John L. (1975), General Topology, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90125-1,MR 0370454

James Munkres (1999), Topology (2nd ed.), Prentice Hall, ISBN 0-13-181629-2, Zbl 0951.54001 Willard, Stephen (1970),General Topology, Addison-Wesley, ISBN3-88538-006-4,MR0264581, Zbl 0205.26601

Chapter 3

Appert topology

In general topology, a branch of mathematics, the Appert topology, named for Appert (1934), is an example of atopology on the set Z+ = {1, 2, 3, } of positive integers.[1] To give Z+ a topology means to say which subsets of Z+are open in a manner that satises certain axioms:[2]

1. The union of open sets is an open set.2. The nite intersection of open sets is an open set.3. Z+ and the empty set are open sets.

In the Appert topology, the open sets are those that do not contain 1, and those that asymptotically contain almostevery positive integer.

3.1 ConstructionLet S be a subset of Z+, and let N(n,S) denote the number of elements of S which are less than or equal to n:

N(n; S) = #fm 2 S : m ng:

In Apperts topology, a set S is dened to be open if either it does not contain 1 or N(n,S)/n tends towards 1 as n tendstowards innity:[1]

limn!1

N(n; S)n

= 1:

The empty set is an open set in this topology because is a set that does not contain 1, and the whole set Z+ is alsoopen in this topology since

Nn;Z+

= n ;

meaning that N(n,S)/n = 1 for all n.

3.2 Related topologiesThe Appert topology is closely related to the Fort space topology that arises from giving the set of integers greaterthan one the discrete topology, and then taking the point 1 as the point at innity in a one point compactication ofthe space.[1] The Fort space is a renement of the Appert topology.

5

6 CHAPTER 3. APPERT TOPOLOGY

3.3 PropertiesThe closed subsets of Z+, equipped with the Appert topology, are the subsets S that either contain 1 or for which

limn!1

N(n; S)n

= 0:

As a result, Z+ is a completely normal space (and thus also Hausdor), for suppose that A and B are disjoint closedsets. If A B did not contain 1, then A and B would also be open and thus completely separated. On the other hand,if A contains 1 then B is open and limn!1 N(n;B)/n=0 , so that Z+B is an open neighborhood of A disjoint from B.[1]

A subset of Z+ is compact in the Appert topology if and only if it is nite. In particular, Z+ is not locally compact,since there is no compact neighborhood of 1. Moreover, Z+ is not countably compact.[1]

3.4 Notes[1] Steen & Seebach 1995, pp. 117118

[2] Steen & Seebach 1995, p. 3

3.5 References Appert, Q (1934), Proprits des Espaces Abstraits les Plus Gnraux, Actual. Sci. Ind. (146), Hermann. Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X.

Chapter 4

Axiom of countability

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

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

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

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

second-countable space: the topology has a countable base

separable space: there exists a countable dense subspace

Lindelf space: every open cover has a countable subcover

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

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

Every rst countable space is sequential.

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

Every -compact space is Lindelf.

Every metric space is rst countable.

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

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

7

8 CHAPTER 4. AXIOM OF COUNTABILITY

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

9780080933795.

Chapter 5

Baire category theorem

The Baire category theorem (BCT) is an important tool in general topology and functional analysis. The theoremhas two forms, each of which gives sucient conditions for a topological space to be a Baire space.The theorem was proved by Ren-Louis Baire in his 1899 doctoral thesis.

5.1 Statement of the theorem

A Baire space is a topological space with the following property: for each countable collection of open dense sets Un,their intersection Un is dense.

(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. Thus every completelymetrizable topological space is a Baire space.

(BCT2) Every locally compact Hausdor space is a Baire space. The proof is similar to the preceding state-ment; the nite intersection property takes the role played by completeness.

Note that neither of these statements implies the other, since there are complete metric spaces which are not locallycompact (the irrational numbers with the metric dened below; also, any Banach space of innite dimension), andthere are locally compact Hausdor spaces which are not metrizable (for instance, any uncountable product of non-trivial compact Hausdor spaces is such; also, several function spaces used in Functional Analysis; the uncountableFort space). See Steen and Seebach in the references below.

(BCT3) A non-empty complete metric space is NOT the countable union of nowhere-dense closed sets.

This formulation is equivalent to BCT1 and is sometimes more useful in applications. Also: if a non-empty completemetric space is the countable union of closed sets, then one of these closed sets has non-empty interior.

5.2 Relation to the axiom of choice

The proofs of BCT1 and BCT2 for arbitrary complete metric spaces require some form of the axiom of choice; andin fact BCT1 is equivalent over ZF to a weak form of the axiom of choice called the axiom of dependent choices.[1]

A restricted form of the Baire category theorem, in which the complete metric space is also assumed to be separable,is provable in ZF with no additional choice principles.[2] This restricted form applies in particular to the real line, theBaire space , the Cantor space 2, and a separable Hilbert space such as L2(R n).

9

10 CHAPTER 5. BAIRE CATEGORY THEOREM

5.3 Uses of the theoremBCT1 is used in functional analysis to prove the open mapping theorem, the closed graph theorem and the uniformboundedness principle.BCT1 also shows that every complete metric space with no isolated points is uncountable. (If X is a countablecomplete metric space with no isolated points, then each singleton {x} in X is nowhere dense, and so X is of rstcategory in itself.) In particular, this proves that the set of all real numbers is uncountable.BCT1 shows that each of the following is a Baire space:

The space R of real numbers The irrational numbers, with the metric dened by d(x; y) = 1n+1 , where n is the rst index for which thecontinued fraction expansions of x and y dier (this is a complete metric space)

The Cantor set

By BCT2, every nite-dimensional Hausdor manifold is a Baire space, since it is locally compact and Hausdor.This is so even for non-paracompact (hence nonmetrizable) manifolds such as the long line.

5.4 ProofThe following is a standard proof that a complete pseudometric space X is a Baire space.Let Un be a countable collection of open dense subsets. We want to show that the intersection TUn is dense. A subsetis dense if and only if every nonempty open subset intersects it. Thus, to show that the intersection is dense, it issucient to show that any nonempty open set W in X has a point x in common with all of the Un . Since U1 is dense,W intersects U1 ; thus, there is a point x1 and 0 < r1 < 1 such that:

B(x1; r1) W \ U1where B(x;r) and B(x;r) denote an open and closed ball, respectively, centered at x with radius r . Since each Un isdense, we can continue recursively to nd a pair of sequences xn and 0 < rn < 1n such that:

B(xn; rn) B(xn1; rn1) \ Un(This step relies on the axiom of choice.) Since xn 2 B(xm;rm) when n > m , we have that xn is Cauchy, and hencexn converges to some limit x by completeness. For any n , by closedness,

x 2 B(xn; rn):Therefore x 2 W and x 2 Un for all n .See also this blog post by M. Baker for the proof of the theorem using Choquets game.

5.5 See also Property of Baire

5.6 Notes[1] Blair 1977

[2] Levy 1979, p. 212

5.7. REFERENCES 11

5.7 References R. Baire. Sur les fonctions de variables relles. Ann. di Mat., 3:1123, 1899. Blair, Charles E. (1977), The Baire category theorem implies the principle of dependent choices., Bull. Acad.

Polon. Sci. Sr. Sci. Math. Astronom. Phys., v. 25 n. 10, pp. 933934.

Levy, Azriel (1979), Basic Set Theory. Reprinted by Dover, 2002. ISBN 0-486-42079-5 Schechter, Eric, Handbook of Analysis and its Foundations, Academic Press, ISBN 0-12-622760-8 Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology, Springer-Verlag, New York,1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).

5.8 External links T. Tao, 245B, Notes 9: The Baire category theorem and its Banach space consequences Encyclopaedia of Mathematics article on Baire theorem

Chapter 6

Baire function

In mathematics, Baire functions are functions obtained from continuous functions by transnite iteration of theoperation of forming pointwise limits of sequences of functions. They were introduced by Ren-Louis Baire (1905).A Baire set is a set whose characteristic function is a Baire function (not necessarily of any particular class, as denedbelow).

6.1 Classication of Baire functionsBaire functions of class n, for any countable ordinal number n, form a vector space of real-valued functions denedon a topological space, as follows.

The Baire class 0 functions are the continuous functions.

The Baire class 1 functions are those functions which are the pointwise limit of a sequence of Baire class 0functions.

In general, the Baire class n functions are all functions which are the pointwise limit of a sequence of functionsof Baire class less than n.

Some authors dene the classes slightly dierently, by removing all functions of class less than n from the functionsof class n. This means that each Baire function has a well dened class, but the functions of given class no longerform a vector space.Henri Lebesgue proved that (for functions on the unit interval) each Baire class of a countable ordinal number containsfunctions not in any smaller class, and that there exist functions which are not in any Baire class.

6.2 Baire class 1Examples:

The derivative of any dierentiable function is of class 1. An example of a dierentiable function whosederivative is not continuous (at x=0) is the function equal to x2 sin(1/x) when x0, and 0 when x=0. Aninnite sum of similar functions (scaled and displaced by rational numbers) can even give a dierentiablefunction whose derivative is discontinuous on a dense set. However, it necessarily has points of continuity,which follows easily from The Baire Characterisation Theorem (below; take K=X=R).

The function equal to 1 if x is an integer and 0 otherwise. (An innite number of large discontinuities.) The function that is 0 for irrational x and 1/q for a rational number p/q (in reduced form). (A dense set ofdiscontinuities, namely the set of rational numbers.)

12

6.3. BAIRE CLASS 2 13

The characteristic function of the Cantor set, which gives 1 if x is in the Cantor set and 0 otherwise. Thisfunction is 0 for an uncountable set of x values, and 1 for an uncountable set. It is discontinuous whereverit equals 1 and continuous wherever it equals 0. It is approximated by the continuous functions gn(x) =max(0; 1 nd(x;C)) , where d(x;C) is the distance of x from the nearest point in the Cantor set.

The Baire Characterisation Theorem states that a real valued function f dened on a Banach space X is a Baire-1function if and only if for every non-empty closed subset K of X, the restriction of f to K has a point of continuityrelative to the topology of K.By another theorem of Baire, for every Baire-1 function the points of continuity are a comeagerGset (Kechris 1995,Theorem (24.14)).


An example of a Baire class two function on the interval [0,1] that is not of class 1 is the characteristic functionof the rational numbers, Q , also known as the Dirichlet function. It is discontinuous everywhere.


6.5 See also Baire set Nowhere continuous function

6.6 References Baire, R. (1905), Leons sur les fonctions discontinues, professes au collge de France, Gauthier-Villars Kechris, Alexander S. (1995), Classical Descriptive Set Theory, Springer-Verlag

6.7 External links Springer Encyclopaedia of Mathematics article on Baire classes

Chapter 7

Baire set

For sets having the property of Baire, which are sometimes called Baire sets, see Property of Baire.

In mathematics, more specically in measure theory, the Baire sets of a locally compact Hausdor space form a-algebra related to the continuous functions on the space. There are several inequivalent denitions of Baire set,which all coincide for the case of locally compact -compact Hausdor spaces.The Baire sets form a subclass of the Borel sets. The converse holds in many, but not all, topological spaces.Baire sets were introduced by Kunihiko Kodaira (1941, Denition 4), Shizuo Kakutani and Kunihiko Kodaira (1944)and Halmos (1950, page 220), who named them after Baire functions that are in turn named after Ren-Louis Baire.They introduced them to avoid some pathological properties of Borel sets on spaces without a countable base for thetopology. In practice, the use of Baire measures on Baire sets can often be replaced by the use of regular Borelmeasures on Borel sets.

7.1 Basic denitionsThere are at least three inequivalent denitions of Baire sets on locally compact Hausdor spaces, and even moredenitions for general topological spaces, though all these denitions are equivalent for locally compact -compactHausdor spaces. Moreover some authors add restrictions on the topological space that Baire sets are dened on, andonly dene Baire sets on spaces that are compact Hausdor, or locally compact Hausdor, or -compact.

7.1.1 First denition

Kunihiko Kodaira dened [1] what we call Baire sets (although he confusingly calls them Borel sets) of certaintopological spaces to be the sets whose characteristic function is a Baire function (the smallest class of functionscontaining all continuous real valued functions and closed under pointwise limits of sequences). Dudley (1989, Sect.7.1) gives an equivalent denition and denes Baire sets of a topological space to be elements of the smallest -algebrasuch that all continuous functions are measurable. For locally compact -compact Hausdor spaces this is equivalentto the following denitions, but in general the denitions are not equivalent.Conversely the Baire functions are exactly the real-valued functions that are Baire measurable. For metric spaces theBaire sets are the same as Borel sets

7.1.2 Second denition

Halmos (1950, page 220) dened Baire sets of a locally compact Hausdor space to be the elements of the -ringgenerated by the compact G sets. This denition is no longer used much as -rings are somewhat out of fashion.When the space is -compact this denition is equivalent to the next denition.One reason for working with compact G sets rather than closed G sets is that Baire measures are then automaticallyregular (Halmos 1950, theorem G page 228).

14

7.2. EXAMPLES 15

7.1.3 Third denition

This is similar to Halmoss denition, modied slightly so that the Baire sets form a -algebra rather than just a -ring.A subset of a locally compact Hausdor topological space is called a Baire set if it is a member of the smallestalgebra containing all compact G sets. In other words, the algebra of Baire sets is the algebra generatedby all compact G sets. Alternatively Baire sets form the smallest -algebra such that all continuous functions ofcompact support are measurable (at least on locally compact Hausdor spaces: on general topological spaces thesetwo conditions need not be equivalent).For -compact spaces this is equivalent to Halmoss denition. For spaces that are not -compact the Baire setsunder this denition are those under Halmoss denition together with their complements. However in this case itis no longer true that a nite Baire measure is necessarily regular: for example the Baire probability measure thatassigns measure 0 to every countable subset of an uncountable discrete space and measure 1 to every co-countablesubset is a Baire probability measure that is not regular.

7.2 Examples

7.2.1 The dierent denitions of Baire sets are not equivalent

For locally compact Hausdor topological spaces that are not -compact the three denitions above need not beequivalent,A discrete topological space is locally compact and Hausdor. Any function dened on a discrete space is continuous,and therefore, according to the rst denition, all subsets of a discrete space are Baire. However, since the compactsubspaces of a discrete space are precisely the nite subspaces, the Baire sets, according to the second denition, areprecisely the at most countable sets, while according to the third denition the Baire sets are the at most countablesets and their complements. Thus, the three denitions are non-equivalent on an uncountable discrete space.For non-Hausdor spaces the denitions of Baire sets in terms of continuous functions need not be equivalent todenitions involving G compact sets. For example, if X is an innite countable set whose closed sets are the nitesets and the whole space, then the only continuous real functions on X are constant, but all subsets of X are in the-algebra generated by compact closed G sets.

7.2.2 A Borel set that is not a Baire set

In a Cartesian product of uncountably many compact Hausdor spaces with more than one point, a point is never aBaire set, in spite of the fact that it is closed, and therefore a Borel set.[2]

7.3 PropertiesBaire sets coincide with Borel sets in Euclidean spaces.For every compact Hausdor space, every nite Baire measure (that is, a measure on the -algebra of all Baire sets)is regular.[3]

For every compact Hausdor space, every nite Baire measure has a unique extension to a regular Borel measure.[4]

The Kolmogorov extension theorem states that every consistent collection of nite-dimensional probability distribu-tions leads to a Baire measure on the space of functions.[5] Assuming compactness one may extend it to a regularBorel measure. After completion one gets a probability space that is not necessarily standard.[6]

7.4 Notes[1] Kodaira 1941, p. 21, Def. 4

[2] Dudley 1989, Example after Theorem 7.1.1

16 CHAPTER 7. BAIRE SET

[3] Dudley 1989, Theorem 7.1.5



[6] Its standardness is investigated in: Tsirelson, Boris (1981). A natural modication of a random process and its applicationto stochastic functional series andGaussianmeasures. Journal of SovietMathematics 16 (2): 940956. doi:10.1007/BF01676139..See Theorem 1(c).

7.5 References Halmos, P. R. (1950). Measure theory. v. Nostrand. See especially Sect. 51 Borel sets and Baire sets. Dudley, R. M. (1989). Real Analysis and Probability. Chapman & Hall. ISBN 0521007542.. See especiallySect. 7.1 Baire and Borel algebras and regularity of measures and Sect. 7.3 The regularity extension.

Kakutani, Shizuo; Kodaira, Kunihiko (1944), "ber das Haarsche Mass in der lokal bikompakten Gruppe,Proc. Imp. Acad. Tokyo 20: 444450, doi:10.3792/pia/1195572875, MR 0014401

Kodaira, Kunihiko (1941), "ber die Gruppe der messbaren Abbildungen, Proc. Imp. Acad. Tokyo 17:1823, doi:10.3792/pia/1195578914, MR 0004089

Chapter 8

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.

8.1 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.

8.2 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.

8.2.1 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.

8.2.2 Historical denitionMain article: Meagre set

17

18 CHAPTER 8. BAIRE SPACE

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.

8.3 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.

8.4 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.

8.5. PROPERTIES 19

8.5 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.

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

8.7 References

8.8 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.

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

Chapter 9

BanachMazur game

In general topology, set theory and game theory, aBanachMazur game is a topological game played by two players,trying to pin down elements in a set (space). The concept of a BanachMazur game is closely related to the concept ofBaire spaces. This game was the rst innite positional game of perfect information to be studied. It was introducedby Mazur as problem 43 in the Scottish book, and Mazurs questions about it were answered by Banach.

9.1 Denition and propertiesIn what follows we will make use of the formalism dened in Topological game. A general BanachMazur game isdened as follows: we have a topological space Y , a xed subset X Y , and a family W of subsets of Y thatsatisfy the following properties.

Each member ofW has non-empty interior. Each non-empty open subset of Y contains a member ofW .

We will call this gameMB(X;Y;W ) . Two players, P1 and P2 , choose alternatively elementsW0 ,W1 , ofWsuch thatW0 W1 . The player P1 wins if and only ifX \ (\n P2 " MB(X;Y;W ) if and only if X is of the rst category in Y (a set is of the rst category or meagre if itis the countable union of nowhere-dense sets).

Assuming that Y is a complete metric space, P1 " MS(X;Y;W ) if and only if X is comeager in somenonempty open subset of Y .

If X has the Baire property in Y , thenMB(X;Y;W ) is determined. Any winning strategy of P2 can be reduced to a stationary winning strategy. The siftable and strongly-siftable spaces introduced by Choquet can be dened in terms of stationary strategiesin suitable modications of the game. Let BM(X) denote a modication ofMB(X;Y;W ) where X = Y ,W is the family of all nonempty open sets inX , andP2 wins a play (W0;W1; ) if and only if\n A Markov winning strategy for P2 in BM(X) can be reduced to a stationary winning strategy. Furthermore,if P2 has a winning strategy in BM(X) , then she has a winning strategy depending only on two precedingmoves. It is still an unsettled question whether a winning strategy for P2 can be reduced to a winning strategythat depends only on the last two moves of P1 .

X is called weakly -favorable if P2 has a winning strategy in BM(X) . Then, X is a Baire space if andonly if P1 has no winning strategy inBM(X) . It follows that each weakly -favorable space is a Baire space.

20

9.2. A SIMPLE PROOF: WINNING STRATEGIES 21

Many other modications and specializations of the basic game have been proposed: for a thorough account of these,refer to [1987]. The most common special case, calledMB(X; J) , consists in letting Y = J , i.e. the unit interval[0; 1] , and in letting W consist of all closed intervals [a; b] contained in [0; 1] . The players choose alternativelysubintervals J0; J1; of J such that J0 J1 , and P1 wins if and only if X \ (\n9.2 A simple proof: winning strategiesIt is natural to ask for what setsX does P2 have a winning strategy. Clearly, ifX is empty, P2 has a winning strategy,therefore the question can be informally rephrased as how small (respectively, big) does X (respectively, thecomplement ofX in Y ) have to be to ensure that P2 has a winning strategy. To give a avor of how the proofs usedto derive the properties in the previous section work,

Documents

Dense set