Hausdorff spaceFrom Wikipedia, the free encyclopedia

Contents

1 Closed set 11.1 Equivalent definitions of a closed set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Properties of closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Examples of closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 More about closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Empty set 32.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 Operations on the empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 In other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Extended real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Questioned existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.1 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.2 Philosophical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 General topology 93.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 A topology on a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.1 Basis for a topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.2 Subspace and quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.3 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3.1 Alternative definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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3.3.3 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.4 Defining topologies via continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4 Compact sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.5 Connected sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.5.1 Connected components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5.2 Disconnected spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5.3 Path-connected sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.6 Products of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.7 Separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.8 Countability axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.9 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.10 Baire category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.11 Main areas of research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.11.1 Continuum theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.11.2 Pointless topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.11.3 Dimension theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.11.4 Topological algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.11.5 Metrizability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.11.6 Set-theoretic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.12 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.14 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Hausdorff space 224.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 Preregularity versus regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.6 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.7 Algebra of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.8 Academic humour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Neighbourhood (mathematics) 265.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 In a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.4 Topology from neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.5 Uniform neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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5.6 Deleted neighbourhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Open set 316.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.2.1 Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.2.2 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.2.3 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.4 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.5 Notes and cautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6.5.1 “Open” is defined relative to a particular topology . . . . . . . . . . . . . . . . . . . . . . 346.5.2 Open and closed are not mutually exclusive . . . . . . . . . . . . . . . . . . . . . . . . . 34

6.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7 Separable space 367.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.2 Separability versus second countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.3 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.4 Constructive mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.5 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.5.1 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.5.2 Non-separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.6.1 Embedding separable metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

8 Subset 408.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.2 ⊂ and ⊃ symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9 Topological space 449.1 History of Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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9.2.1 Neighbourhoods definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.2.2 Open sets definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.2.3 Closed sets definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.2.4 Other definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

9.3 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.4 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.5 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.6 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.7 Classification of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.8 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.9 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.10 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

10 Topology 5110.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5210.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5310.3 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

10.3.1 Topologies on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.3.2 Continuous functions and homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 5610.3.3 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

10.4 Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.4.1 General topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.4.2 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.4.3 Differential topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.4.4 Geometric topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.4.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

10.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5810.5.1 Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5810.5.2 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5810.5.3 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5810.5.4 Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

10.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5810.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5910.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6010.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6010.10Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 61

10.10.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6110.10.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

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10.10.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Chapter 1

Closed set

This article is about the complement of an open set. For a set closed under an operation, see closure (mathematics).For other uses, see Closed (disambiguation).

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an openset.[1][2] In a topological space, a closed set can be defined as a set which contains all its limit points. In a completemetric space, a closed set is a set which is closed under the limit operation.

1.1 Equivalent definitions of a closed set

In a topological space, a set is closed if and only if it coincides with its closure. Equivalently, a set is closed if andonly if it contains all of its limit points.This is not to be confused with a closed manifold.

1.2 Properties of closed sets

A closed set contains its own boundary. In other words, if you are “outside” a closed set, you may move a smallamount in any direction and still stay outside the set. Note that this is also true if the boundary is the empty set, e.g.in the metric space of rational numbers, for the set of numbers of which the square is less than 2.

• Any intersection of closed sets is closed (including intersections of infinitely many closed sets)

• The union of finitely many closed sets is closed.

• The empty set is closed.

• The whole set is closed.

In fact, given a set X and a collection F of subsets of X that has these properties, then F will be the collection ofclosed sets for a unique topology on X. The intersection property also allows one to define the closure of a set A in aspace X, which is defined as the smallest closed subset of X that is a superset of A. Specifically, the closure of A canbe constructed as the intersection of all of these closed supersets.Sets that can be constructed as the union of countably many closed sets are denoted Fσ sets. These sets need not beclosed.

1.3 Examples of closed sets• The closed interval [a,b] of real numbers is closed. (See Interval (mathematics) for an explanation of thebracket and parenthesis set notation.)

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2 CHAPTER 1. CLOSED SET

• The unit interval [0,1] is closed in the metric space of real numbers, and the set [0,1] ∩ Q of rational numbersbetween 0 and 1 (inclusive) is closed in the space of rational numbers, but [0,1] ∩ Q is not closed in the realnumbers.

• Some sets are neither open nor closed, for instance the half-open interval [0,1) in the real numbers.• Some sets are both open and closed and are called clopen sets.• Half-interval [1, +∞) is closed.• The Cantor set is an unusual closed set in the sense that it consists entirely of boundary points and is nowheredense.

• Singleton points (and thus finite sets) are closed in Hausdorff spaces.• If X and Y are topological spaces, a function f from X into Y is continuous if and only if preimages of closedsets in Y are closed in X.

1.4 More about closed sets

In point set topology, a set A is closed if it contains all its boundary points.The notion of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces,as well as for other spaces that carry topological structures, such as metric spaces, differentiable manifolds, uniformspaces, and gauge spaces.An alternative characterization of closed sets is available via sequences and nets. A subset A of a topological spaceX is closed in X if and only if every limit of every net of elements of A also belongs to A. In a first-countable space(such as a metric space), it is enough to consider only convergent sequences, instead of all nets. One value of thischaracterisation is that it may be used as a definition in the context of convergence spaces, which are more generalthan topological spaces. Notice that this characterisation also depends on the surrounding space X, because whetheror not a sequence or net converges in X depends on what points are present in X.Whether a set is closed depends on the space in which it is embedded. However, the compact Hausdorff spacesare "absolutely closed", in the sense that, if you embed a compact Hausdorff space K in an arbitrary Hausdorffspace X, then K will always be a closed subset of X; the “surrounding space” does not matter here. Stone-Čechcompactification, a process that turns a completely regular Hausdorff space into a compact Hausdorff space, may bedescribed as adjoining limits of certain nonconvergent nets to the space.Furthermore, every closed subset of a compact space is compact, and every compact subspace of a Hausdorff spaceis closed.Closed sets also give a useful characterization of compactness: a topological space X is compact if and only ifevery collection of nonempty closed subsets of X with empty intersection admits a finite subcollection with emptyintersection.A topological space X is disconnected if there exist disjoint, nonempty, closed subsets A and B of X whose union isX. Furthermore, X is totally disconnected if it has an open basis consisting of closed sets.

1.5 See also• Open set• Clopen set• Neighbourhood

1.6 References[1] Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill. ISBN 0-07-054235-X.

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

Chapter 2

Empty set

"∅" redirects here. For similar symbols, see Ø (disambiguation).In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size orcardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists byincluding an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of setsare trivially true for the empty set.Null set was once a common synonym for “empty set”, but is now a technical term in measure theory. The empty setmay also be called the void set.

2.1 Notation

Common notations for the empty set include "{}", "∅", and " ∅ ". The latter two symbols were introduced by theBourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Norwegian and Danish alphabets(and not related in any way to the Greek letter Φ).[1]

The empty-set symbol ∅ is found at Unicode point U+2205.[2] In TeX, it is coded as \emptyset or \varnothing.

2.2 Properties

In standard axiomatic set theory, by the principle of extensionality, two sets are equal if they have the same elements;therefore there can be only one set with no elements. Hence there is but one empty set, and we speak of “the emptyset” rather than “an empty set”.The mathematical symbols employed below are explained here.For any set A:

• The empty set is a subset of A:

∀A : ∅ ⊆ A

• The union of A with the empty set is A:

∀A : A ∪ ∅ = A

• The intersection of A with the empty set is the empty set:

∀A : A ∩ ∅ = ∅

• The Cartesian product of A and the empty set is the empty set:

∀A : A× ∅ = ∅

3

4 CHAPTER 2. EMPTY SET

The empty set is the set containing no elements.

The empty set has the following properties:

• Its only subset is the empty set itself:

∀A : A ⊆ ∅ ⇒ A = ∅

• The power set of the empty set is the set containing only the empty set:

2∅ = {∅}

2.2. PROPERTIES 5

A symbol for the empty set

• Its number of elements (that is, its cardinality) is zero:

card(∅) = 0

The connection between the empty set and zero goes further, however: in the standard set-theoretic definition ofnatural numbers, we use sets to model the natural numbers. In this context, zero is modelled by the empty set.For any property:

• For every element of ∅ the property holds (vacuous truth);

• There is no element of ∅ for which the property holds.

Conversely, if for some property and some set V, the following two statements hold:

• For every element of V the property holds;

• There is no element of V for which the property holds,

V = ∅

6 CHAPTER 2. EMPTY SET

By the definition of subset, the empty set is a subset of any set A, as every element x of ∅ belongs to A. If it is nottrue that every element of ∅ is in A, there must be at least one element of ∅ that is not present in A. Since there areno elements of ∅ at all, there is no element of ∅ that is not in A. Hence every element of ∅ is in A, and ∅ is a subsetof A. Any statement that begins “for every element of ∅ " is not making any substantive claim; it is a vacuous truth.This is often paraphrased as “everything is true of the elements of the empty set.”

2.2.1 Operations on the empty set

Operations performed on the empty set (as a set of things to be operated upon) are unusual. For example, the sumof the elements of the empty set is zero, but the product of the elements of the empty set is one (see empty product).Ultimately, the results of these operations say more about the operation in question than about the empty set. Forinstance, zero is the identity element for addition, and one is the identity element for multiplication.A disarrangement of a set is a permutation of the set that leaves no element in the same position. The empty set is adisarrangment of itself as no element can be found that retains its original position.

2.3 In other areas of mathematics

2.3.1 Extended real numbers

Since the empty set has no members, when it is considered as a subset of any ordered set, then every member ofthat set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of thereal numbers, with its usual ordering, represented by the real number line, every real number is both an upper andlower bound for the empty set.[3] When considered as a subset of the extended reals formed by adding two “numbers”or “points” to the real numbers, namely negative infinity, denoted −∞, which is defined to be less than every otherextended real number, and positive infinity, denoted +∞, which is defined to be greater than every other extendedreal number, then:

sup ∅ = min({−∞,+∞} ∪ R) = −∞,

and

inf ∅ = max({−∞,+∞} ∪ R) = +∞.

That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound(inf or infimum) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinityis the identity element for the maximum and supremum operators, while positive infinity is the identity element forminimum and infimum.

2.3.2 Topology

Considered as a subset of the real number line (or more generally any topological space), the empty set is both closedand open; it is an example of a “clopen” set. All its boundary points (of which there are none) are in the empty set,and the set is therefore closed; while for every one of its points (of which there are again none), there is an openneighbourhood in the empty set, and the set is therefore open. Moreover, the empty set is a compact set by the factthat every finite set is compact.The closure of the empty set is empty. This is known as “preservation of nullary unions.”

2.3.3 Category theory

If A is a set, then there exists precisely one function f from {} to A, the empty function. As a result, the empty set isthe unique initial object of the category of sets and functions.

2.4. QUESTIONED EXISTENCE 7

The empty set can be turned into a topological space, called the empty space, in just one way: by defining the emptyset to be open. This empty topological space is the unique initial object in the category of topological spaces withcontinuous maps.The empty set is more ever a strict initial object: only the empty set has a function to the empty set.

2.4 Questioned existence

2.4.1 Axiomatic set theory

In Zermelo set theory, the existence of the empty set is assured by the axiom of empty set, and its uniqueness followsfrom the axiom of extensionality. However, the axiom of empty set can be shown redundant in either of two ways:

• There is already an axiom implying the existence of at least one set. Given such an axiom together with theaxiom of separation, the existence of the empty set is easily proved.

• In the presence of urelements, it is easy to prove that at least one set exists, viz. the set of all urelements. Again,given the axiom of separation, the empty set is easily proved.

2.4.2 Philosophical issues

While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity,whose meaning and usefulness are debated by philosophers and logicians.The empty set is not the same thing as nothing; rather, it is a set with nothing inside it and a set is always something.This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. Darling (2004) explainsthat the empty set is not nothing, but rather “the set of all triangles with four sides, the set of all numbers that arebigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king.”[4]

The popular syllogism

Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sand-wich is better than eternal happiness

is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darlingwrites that the contrast can be seen by rewriting the statements “Nothing is better than eternal happiness” and "[A]ham sandwich is better than nothing” in a mathematical tone. According to Darling, the former is equivalent to “Theset of all things that are better than eternal happiness is ∅ " and the latter to “The set {ham sandwich} is better thanthe set ∅ ". It is noted that the first compares elements of sets, while the second compares the sets themselves.[4]

Jonathan Lowe argues that while the empty set:

"...was undoubtedly an important landmark in the history of mathematics, … we should not assume thatits utility in calculation is dependent upon its actually denoting some object.”

it is also the case that:

“All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and(3) is unique amongst sets in having no members. However, there are very many things that 'have nomembers’, in the set-theoretical sense—namely, all non-sets. It is perfectly clear why these things haveno members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a setwhich has no members. We cannot conjure such an entity into existence by mere stipulation.”[5]

George Boolos argued that much of what has been heretofore obtained by set theory can just as easily be obtainedby plural quantification over individuals, without reifying sets as singular entities having other entities as members.[6]

8 CHAPTER 2. EMPTY SET

2.5 See also• Inhabited set

• Nothing

2.6 Notes[1] Earliest Uses of Symbols of Set Theory and Logic.

[2] Unicode Standard 5.2

[3] Bruckner, A.N., Bruckner, J.B., and Thomson, B.S., 2008. Elementary Real Analysis, 2nd ed. Prentice Hall. P. 9.

[4] D. J. Darling (2004). The universal book of mathematics. John Wiley and Sons. p. 106. ISBN 0-471-27047-4.

[5] E. J. Lowe (2005). Locke. Routledge. p. 87.

[6] • George Boolos, 1984, “To be is to be the value of a variable,” The Journal of Philosophy 91: 430–49. Reprinted inhis 1998 Logic, Logic and Logic (Richard Jeffrey, and Burgess, J., eds.) Harvard Univ. Press: 54–72.

2.7 References• Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books,2011. ISBN 978-1-61427-131-4 (Paperback edition).

• Jech, Thomas (2002), Set Theory, Springer Monographs in Mathematics (3rd millennium ed.), Springer, ISBN3-540-44085-2

• Graham,Malcolm (1975),Modern ElementaryMathematics (Hardcover) (2nd ed.), NewYork: Harcourt BraceJovanovich, ISBN 0155610392

2.8 External links• Weisstein, Eric W., “Empty Set”, MathWorld.

Chapter 3

General topology

The Topologist’s sine curve, a useful example in point-set topology. It is connected but not path-connected.

In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions andconstructions used in topology. It is the foundation ofmost other branches of topology, including differential topology,geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness:

• Continuous functions, intuitively, take nearby points to nearby points.

• Compact sets are those that can be covered by finitely many sets of arbitrarily small size.

• Connected sets are sets that cannot be divided into two pieces that are far apart.

The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below.If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are.Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.

9

10 CHAPTER 3. GENERAL TOPOLOGY

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

3.1 History

General topology grew out of a number of areas, most importantly the following:

• the detailed study of subsets of the real line (once known as the topology of point sets, this usage is now obsolete)

• the introduction of the manifold concept

• the study of metric spaces, especially normed linear spaces, in the early days of functional analysis.

General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuitionof continuity, in a technically adequate form that can be applied in any area of mathematics.

3.2 A topology on a set

Main article: Topological space

Let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:[1][2]

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

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

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

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

3.2.1 Basis for a topology

Main article: Basis (topology)

A base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open setin T can be written as a union of elements of B.[3][4] We say that the base generates the topology T. Bases are usefulbecause many properties of topologies can be reduced to statements about a base that generates that topology—andbecause many topologies are most easily defined in terms of a base that generates them.

3.2.2 Subspace and quotient

Every subset of a topological space can be given the subspace topology in which the open sets are the intersectionsof the open sets of the larger space with the subset. For any indexed family of topological spaces, the product can begiven the product topology, which is generated by the inverse images of open sets of the factors under the projectionmappings. For example, in finite products, a basis for the product topology consists of all products of open sets. Forinfinite products, there is the additional requirement that in a basic open set, all but finitely many of its projectionsare the entire space.A quotient space is defined as follows: if X is a topological space and Y is a set, and if f : X→ Y is a surjectivefunction, then the quotient topology on Y is the collection of subsets of Y that have open inverse images under f. In

3.3. CONTINUOUS FUNCTIONS 11

other words, the quotient topology is the finest topology on Y for which f is continuous. A common example of aquotient topology is when an equivalence relation is defined on the topological space X. The map f is then the naturalprojection onto the set of equivalence classes.

3.2.3 Examples of topological spaces

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

3.3 Continuous functions

Main article: Continuous function

12 CHAPTER 3. GENERAL TOPOLOGY

Continuity is expressed in terms of neighborhoods: f is continuous at some point x ∈ X if and only if for any neigh-borhood V of f(x), there is a neighborhood U of x such that f(U) ⊆ V. Intuitively, continuity means no matter how“small” V becomes, there is always a U containing x that maps inside V and whose image under f contains f(x).This is equivalent to the condition that the preimages of the open (closed) sets in Y are open (closed) in X. In metricspaces, this definition is equivalent to the ε–δ-definition that is often used in analysis.An extreme example: if a set X is given the discrete topology, all functions

f : X → T

to any topological space T are continuous. On the other hand, if X is equipped with the indiscrete topology and thespace T set is at least T0, then the only continuous functions are the constant functions. Conversely, any functionwhose range is indiscrete is continuous.

3.3.1 Alternative definitions

Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define acontinuous function.

Neighborhood definition

Definitions based on preimages are often difficult to use directly. The following criterion expresses continuity interms of neighborhoods: f is continuous at some point x ∈ X if and only if for any neighborhood V of f(x), there isa neighborhood U of x such that f(U) ⊆ V. Intuitively, continuity means no matter how “small” V becomes, there isalways a U containing x that maps inside V.If X and Y are metric spaces, it is equivalent to consider the neighborhood system of open balls centered at x and f(x)instead of all neighborhoods. This gives back the above δ-ε definition of continuity in the context of metric spaces.However, in general topological spaces, there is no notion of nearness or distance.Note, however, that if the target space is Hausdorff, it is still true that f is continuous at a if and only if the limit off as x approaches a is f(a). At an isolated point, every function is continuous.

Sequences and nets

In several contexts, the topology of a space is conveniently specified in terms of limit points. In many instances, thisis accomplished by specifying when a point is the limit of a sequence, but for some spaces that are too large in somesense, one specifies also when a point is the limit of more general sets of points indexed by a directed set, knownas nets.[5] A function is continuous only if it takes limits of sequences to limits of sequences. In the former case,preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to becontinuous, and preservation of nets is a necessary and sufficient condition.In detail, a function f: X → Y is sequentially continuous if whenever a sequence (xn) in X converges to a limit x,the sequence (f(xn)) converges to f(x).[6] Thus sequentially continuous functions “preserve sequential limits”. Everycontinuous function is sequentially continuous. If X is a first-countable space and countable choice holds, then theconverse also holds: any function preserving sequential limits is continuous. In particular, if X is a metric space,sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might bestrictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.)This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functionspreserve limits of nets, and in fact this property characterizes continuous functions.

Closure operator definition

Instead of specifying the open subsets of a topological space, the topology can also be determined by a closure operator(denoted cl), which assigns to any subset A ⊆ X its closure, or an interior operator (denoted int), which assigns to anysubset A of X its interior. In these terms, a function

3.3. CONTINUOUS FUNCTIONS 13

f : (X, cl) → (X ′, cl′)

between topological spaces is continuous in the sense above if and only if for all subsets A of X

f(cl(A)) ⊆ cl′(f(A)).

That is to say, given any element x of X that is in the closure of any subset A, f(x) belongs to the closure of f(A).This is equivalent to the requirement that for all subsets A' of X'

f−1(cl′(A′)) ⊇ cl(f−1(A′)).

Moreover,

f : (X, int) → (X ′, int′)

is continuous if and only if

f−1(int′(A)) ⊆ int(f−1(A))

for any subset A of X.

3.3.2 Properties

If f: X→ Y and g: Y → Z are continuous, then so is the composition g ∘ f: X→ Z. If f: X→ Y is continuous and

• X is compact, then f(X) is compact.

• X is connected, then f(X) is connected.

• X is path-connected, then f(X) is path-connected.

• X is Lindelöf, then f(X) is Lindelöf.

• X is separable, then f(X) is separable.

The possible topologies on a fixed setX are partially ordered: a topology τ1 is said to be coarser than another topologyτ2 (notation: τ1 ⊆ τ2) if every open subset with respect to τ1 is also open with respect to τ2. Then, the identity map

idX: (X, τ2) → (X, τ1)

is continuous if and only if τ1 ⊆ τ2 (see also comparison of topologies). More generally, a continuous function

(X, τX) → (Y, τY )

stays continuous if the topology τY is replaced by a coarser topology and/or τX is replaced by a finer topology.

3.3.3 Homeomorphisms

Symmetric to the concept of a continuous map is an open map, for which images of open sets are open. In fact, if anopen map f has an inverse function, that inverse is continuous, and if a continuous map g has an inverse, that inverseis open. Given a bijective function f between two topological spaces, the inverse function f−1 need not be continuous.A bijective continuous function with continuous inverse function is called a homeomorphism.If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is a homeomorphism.

14 CHAPTER 3. GENERAL TOPOLOGY

3.3.4 Defining topologies via continuous functions

Given a function

f : X → S,

where X is a topological space and S is a set (without a specified topology), the final topology on S is defined by lettingthe open sets of S be those subsets A of S for which f−1(A) is open in X. If S has an existing topology, f is continuouswith respect to this topology if and only if the existing topology is coarser than the final topology on S. Thus the finaltopology can be characterized as the finest topology on S that makes f continuous. If f is surjective, this topology iscanonically identified with the quotient topology under the equivalence relation defined by f.Dually, for a function f from a set S to a topological space, the initial topology on S has as open subsets A of S thosesubsets for which f(A) is open in X. If S has an existing topology, f is continuous with respect to this topology if andonly if the existing topology is finer than the initial topology on S. Thus the initial topology can be characterized asthe coarsest topology on S that makes f continuous. If f is injective, this topology is canonically identified with thesubspace topology of S, viewed as a subset of X.More generally, given a set S, specifying the set of continuous functions

S → X

into all topological spaces X defines a topology. Dually, a similar idea can be applied to maps

X → S.

This is an instance of a universal property.

3.4 Compact sets

Main article: Compact (mathematics)

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

{Uα}α∈A

of open subsets of X such that

X =∪α∈A

Uα,

there is a finite subset J of A such that

X =∪i∈J

Ui.

Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki,use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are bothHausdorff and quasi-compact. A compact set is sometimes referred to as a compactum, plural compacta.Every closed interval in R of finite length is compact. More is true: In Rn, a set is compact if and only if it is closedand bounded. (See Heine–Borel theorem).

3.5. CONNECTED SETS 15

Every continuous image of a compact space is compact.A compact subset of a Hausdorff space is closed.Every continuous bijection from a compact space to a Hausdorff space is necessarily a homeomorphism.Every sequence of points in a compact metric space has a convergent subsequence.Every compact finite-dimensional manifold can be embedded in some Euclidean space Rn.

3.5 Connected sets

Main article: connected space

A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, Xis said to be connected. A subset of a topological space is said to be connected if it is connected under its subspacetopology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article doesnot follow that practice.For a topological space X the following conditions are equivalent:

1. X is connected.

2. X cannot be divided into two disjoint nonempty closed sets.

3. The only subsets of X that are both open and closed (clopen sets) are X and the empty set.

4. The only subsets of X with empty boundary are X and the empty set.

5. X cannot be written as the union of two nonempty separated sets.

6. The only continuous functions from X to {0,1}, the two-point space endowed with the discrete topology, areconstant.

Every interval in R is connected.The continuous image of a connected space is connected.

3.5.1 Connected components

The maximal connected subsets (ordered by inclusion) of a nonempty topological space are called the connectedcomponents of the space. The components of any topological space X form a partition of X: they are disjoint,nonempty, and their union is the whole space. Every component is a closed subset of the original space. It followsthat, in the case where their number is finite, each component is also an open subset. However, if their number isinfinite, this might not be the case; for instance, the connected components of the set of the rational numbers are theone-point sets, which are not open.Let Γx be the connected component of x in a topological space X, and Γ′

x be the intersection of all open-closed setscontaining x (called quasi-component of x.) Then Γx ⊂ Γ′

x where the equality holds if X is compact Hausdorff orlocally connected.

3.5.2 Disconnected spaces

A space in which all components are one-point sets is called totally disconnected. Related to this property, a space Xis called totally separated if, for any two distinct elements x and y of X, there exist disjoint open neighborhoods Uof x and V of y such that X is the union of U and V. Clearly any totally separated space is totally disconnected, butthe converse does not hold. For example, take two copies of the rational numbersQ, and identify them at every pointexcept zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering thetwo copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdorff, and the conditionof being totally separated is strictly stronger than the condition of being Hausdorff.

16 CHAPTER 3. GENERAL TOPOLOGY

3.5.3 Path-connected sets

This subspace of R² is path-connected, because a path can be drawn between any two points in the space.

A path from a point x to a point y in a topological space X is a continuous function f from the unit interval [0,1] toX with f(0) = x and f(1) = y. A path-component of X is an equivalence class of X under the equivalence relation,which makes x equivalent to y if there is a path from x to y. The space X is said to be path-connected (or pathwiseconnected or 0-connected) if there is at most one path-component, i.e. if there is a path joining any two points inX. Again, many authors exclude the empty space.Every path-connected space is connected. The converse is not always true: examples of connected spaces that arenot path-connected include the extended long line L* and the topologist’s sine curve.However, subsets of the real lineR are connected if and only if they are path-connected; these subsets are the intervalsofR. Also, open subsets ofRn orCn are connected if and only if they are path-connected. Additionally, connectednessand path-connectedness are the same for finite topological spaces.

3.6 Products of spaces

Main article: Product topology

Given X such that

X :=∏i∈I

Xi,

is the Cartesian product of the topological spaces Xi, indexed by i ∈ I , and the canonical projections pi : X→ Xi,the product topology on X is defined as the coarsest topology (i.e. the topology with the fewest open sets) for whichall the projections pi are continuous. The product topology is sometimes called the Tychonoff topology.The open sets in the product topology are unions (finite or infinite) of sets of the form

∏i∈I Ui , where each Ui is

open in Xi and Ui ≠ Xi only finitely many times. In particular, for a finite product (in particular, for the product oftwo topological spaces), the products of base elements of the Xi gives a basis for the product

∏i∈I Xi .

3.7. SEPARATION AXIOMS 17

The product topology on X is the topology generated by sets of the form pi−1(U), where i is in I and U is an opensubset of Xi. In other words, the sets {pi−1(U)} form a subbase for the topology on X. A subset of X is open ifand only if it is a (possibly infinite) union of intersections of finitely many sets of the form pi−1(U). The pi−1(U) aresometimes called open cylinders, and their intersections are cylinder sets.In general, the product of the topologies of each Xi forms a basis for what is called the box topology on X. In general,the box topology is finer than the product topology, but for finite products they coincide.Related to compactness is Tychonoff’s theorem: the (arbitrary) product of compact spaces is compact.

3.7 Separation axioms

Main article: Separation axiom

Many of these names have alternative meanings in some of mathematical literature, as explained on History of theseparation axioms; for example, the meanings of “normal” and “T4" are sometimes interchanged, similarly “regular”and “T3", etc. Many of the concepts also have several names; however, the one listed first is always least likely to beambiguous.Most of these axioms have alternative definitions with the samemeaning; the definitions given here fall into a consistentpattern that relates the various notions of separation defined in the previous section. Other possible definitions can befound in the individual articles.In all of the following definitions, X is again a topological space.

• X is T0, or Kolmogorov, if any two distinct points in X are topologically distinguishable. (It is a common themeamong the separation axioms to have one version of an axiom that requires T0 and one version that doesn't.)

• X is T1, or accessible or Fréchet, if any two distinct points in X are separated. Thus, X is T1 if and onlyif it is both T0 and R0. (Though you may say such things as T1 space, Fréchet topology, and Suppose that thetopological spaceX is Fréchet, avoid saying Fréchet space in this context, since there is another entirely differentnotion of Fréchet space in functional analysis.)

• X is Hausdorff, or T2 or separated, if any two distinct points in X are separated by neighbourhoods. Thus, Xis Hausdorff if and only if it is both T0 and R1. A Hausdorff space must also be T1.

• X is T2½, or Urysohn, if any two distinct points in X are separated by closed neighbourhoods. A T₂½ spacemust also be Hausdorff.

• X is regular, or T3, if it is T0 and if given any point x and closed set F in X such that x does not belong to F,they are separated by neighbourhoods. (In fact, in a regular space, any such x and F is also separated by closedneighbourhoods.)

• X is Tychonoff, or T3½, completely T3, or completely regular, if it is T0 and if f, given any point x and closedset F in X such that x does not belong to F, they are separated by a continuous function.

• X is normal, orT4, if it is Hausdorff and if any two disjoint closed subsets ofX are separated by neighbourhoods.(In fact, a space is normal if and only if any two disjoint closed sets can be separated by a continuous function;this is Urysohn’s lemma.)

• X is completely normal, or T5 or completely T4, if it is T1. and if any two separated sets are separated byneighbourhoods. A completely normal space must also be normal.

• X is perfectly normal, or T6 or perfectly T4, if it is T1 and if any two disjoint closed sets are precisely separatedby a continuous function. A perfectly normal Hausdorff space must also be completely normal Hausdorff.

The Tietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspacecan be extended to a continuous map defined on the whole space.

18 CHAPTER 3. GENERAL TOPOLOGY

3.8 Countability axioms

Main article: axiom of countability

An axiom of countability is a property of certain mathematical objects (usually in a category) that requires theexistence of a countable set with certain properties, while without it such sets might not exist.Important countability axioms for topological spaces:

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

• first-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

• Lindelöf space: every open cover has a countable subcover

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

Relations:

• Every first countable space is sequential.

• Every second-countable space is first-countable, separable, and Lindelöf.

• Every σ-compact space is Lindelöf.

• A metric space is first-countable.

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

3.9 Metric spaces

Main article: Metric space

Ametric space[7] is an ordered pair (M,d) whereM is a set and d is a metric onM , i.e., a function

d : M ×M → R

such that for any x, y, z ∈ M , the following holds:

1. d(x, y) ≥ 0 (non-negative),

2. d(x, y) = 0 iff x = y (identity of indiscernibles),

3. d(x, y) = d(y, x) (symmetry) and

4. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality) .

The function d is also called distance function or simply distance. Often, d is omitted and one just writes M for ametric space if it is clear from the context what metric is used.Every metric space is paracompact and Hausdorff, and thus normal.The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.

3.10. BAIRE CATEGORY THEORY 19

3.10 Baire category theory

Main article: Baire category theorem

The Baire category theorem says: If X is a complete metric space or a locally compact Hausdorff space, then theinterior of every union of countably many nowhere dense sets is empty.[8]

Any open subspace of a Baire space is itself a Baire space.

3.11 Main areas of research

Three iterations of a Peano curve construction, whose limit is a space-filling curve. The Peano curve is studied in continuum theory,a branch of general topology.

3.11.1 Continuum theory

Main article: Continuum theory

A continuum (pl continua) is a nonempty compact connected metric space, or less frequently, a compact connectedHausdorff space. Continuum theory is the branch of topology devoted to the study of continua.

3.11.2 Pointless topology

Main article: Pointless topology

Pointless topology (also called point-free or pointfree topology) is an approach to topology that avoids mentioningpoints. The name 'pointless topology' is due to John von Neumann.[9] The ideas of pointless topology are closelyrelated to mereotopologies, in which regions (sets) are treated as foundational without explicit reference to underlyingpoint sets.

3.11.3 Dimension theory

Main article: Dimension theory

Dimension theory is a branch of general topology dealing with dimensional invariants of topological spaces.

3.11.4 Topological algebras

Main article: Topological algebra

20 CHAPTER 3. GENERAL TOPOLOGY

A topological algebra A over a topological field K is a topological vector space together with a continuous multipli-cation

· : A×A −→ A

(a, b) 7−→ a · bthat makes it an algebra over K. A unital associative topological algebra is a topological ring.The term was coined by David van Dantzig; it appears in the title of his doctoral dissertation (1931).

3.11.5 Metrizability theory

Main article: Metrization theorem

In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to ametric space. That is, a topological space (X, τ) is said to be metrizable if there is a metric

d : X ×X → [0,∞)

such that the topology induced by d is τ . Metrization theorems are theorems that give sufficient conditions for atopological space to be metrizable.

3.11.6 Set-theoretic topology

Main article: Set-theoretic topology

Set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questionsthat are independent of Zermelo–Fraenkel set theory(ZFC). A famous problem is the normal Moore space question, aquestion in general topology that was the subject of intense research. The answer to the normal Moore space questionwas eventually proved to be independent of ZFC.

3.12 See also• List of examples in general topology

• Glossary of general topology for detailed definitions

• List of general topology topics for related articles

• Category of topological spaces

3.13 References[1] Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.

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

[3] Merrifield, Richard E.; Simmons, Howard E. (1989). Topological Methods in Chemistry. New York: John Wiley & Sons.p. 16. ISBN 0-471-83817-9. Retrieved 27 July 2012. Definition. A collection B of subsets of a topological space (X,T)is called a basis for T if every open set can be expressed as a union of members of B.

[4] Armstrong, M. A. (1983). Basic Topology. Springer. p. 30. ISBN 0-387-90839-0. Retrieved 13 June 2013. Suppose wehave a topology on a set X, and a collection β of open sets such that every open set is a union of members of β . Then βis called a base for the topology...

3.14. FURTHER READING 21

[5] Moore, E. H.; Smith, H. L. (1922). “A General Theory of Limits”. American Journal of Mathematics 44 (2): 102–121.doi:10.2307/2370388. JSTOR 2370388

[6] Heine, E.. “Die Elemente der Functionenlehre..” Journal für die reine und angewandte Mathematik 74 (1872): 172-188.<http://eudml.org/doc/148175>.

[7] Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, Rendic. Circ. Mat.Palermo 22 (1906) 1–74.

[8] R. Baire. Sur les fonctions de variables réelles. Ann. di Mat., 3:1–123, 1899.

[9] Garrett Birkhoff, VON NEUMANN AND LATTICE THEORY, John Von Neumann 1903-1957, J. C. Oxtoley, B. J. Pettis,American Mathematical Soc., 1958, page 50-5

3.14 Further reading

Some standard books on general topology include:

• Bourbaki, Topologie Générale (General Topology), ISBN 0-387-19374-X.

• John L. Kelley (1955) General Topology, link from Internet Archive, originally published by David Van Nos-trand Company.

• Stephen Willard, General Topology, ISBN 0-486-43479-6.

• James Munkres, Topology, ISBN 0-13-181629-2.

• George F. Simmons, Introduction to Topology and Modern Analysis, ISBN 1-575-24238-9.

• Paul L. Shick, Topology: Point-Set and Geometric, ISBN 0-470-09605-5.

• Ryszard Engelking, General Topology, ISBN 3-88538-006-4.

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

• O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev, Elementary Topology: Textbook in Prob-lems, ISBN 978-0-8218-4506-6.

The arXiv subject code is math.GN.

Chapter 4

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topologicalspace in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on atopological space, the “Hausdorff condition” (T2) is the most frequently used and discussed. It implies the uniquenessof limits of sequences, nets, and filters.Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff’s original definitionof a topological space (in 1914) included the Hausdorff condition as an axiom.

4.1 Definitions

U

x

V

y

The points x and y, separated by their respective neighbourhoods U and V.

Points x and y in a topological space X can be separated by neighbourhoods if there exists a neighbourhood U of xand a neighbourhood V of y such that U and V are disjoint (U ∩ V = ∅). X is a Hausdorff space if any two distinctpoints of X can be separated by neighborhoods. This condition is the third separation axiom (after T0 and T1), whichis why Hausdorff spaces are also called T2 spaces. The name separated space is also used.A related, but weaker, notion is that of a preregular space. X is a preregular space if any two topologically distin-guishable points can be separated by neighbourhoods. Preregular spaces are also called R1 spaces.The relationship between these two conditions is as follows. A topological space is Hausdorff if and only if it is bothpreregular (i.e. topologically distinguishable points are separated by neighbourhoods) and Kolmogorov (i.e. distinctpoints are topologically distinguishable). A topological space is preregular if and only if its Kolmogorov quotient is

22

4.2. EQUIVALENCES 23

Hausdorff.

4.2 Equivalences

For a topological space X, the following are equivalent:

• X is a Hausdorff space.

• Limits of nets in X are unique.[1]

• Limits of filters on X are unique.[2]

• Any singleton set {x} ⊂ X is equal to the intersection of all closed neighbourhoods of x.[3] (A closed neigh-bourhood of x is a closed set that contains an open set containing x.)

• The diagonal Δ = {(x,x) | x ∈ X} is closed as a subset of the product space X × X.

4.3 Examples and counterexamples

Almost all spaces encountered in analysis are Hausdorff; most importantly, the real numbers (under the standardmetric topology on real numbers) are a Hausdorff space. More generally, all metric spaces are Hausdorff. In fact,many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdorff conditionexplicitly stated in their definitions.A simple example of a topology that is T1 but is not Hausdorff is the cofinite topology defined on an infinite set.Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in theconstruction of Hausdorff gauge spaces. Indeed, when analysts run across a non-Hausdorff space, it is still probablyat least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff.In contrast, non-preregular spaces are encountered much more frequently in abstract algebra and algebraic geometry,in particular as the Zariski topology on an algebraic variety or the spectrum of a ring. They also arise in the modeltheory of intuitionistic logic: every complete Heyting algebra is the algebra of open sets of some topological space,but this space need not be preregular, much less Hausdorff.While the existence of unique limits for convergent nets and filters implies that a space is Hausdorff, there are non-Hausdorff T1 spaces in which every convergent sequence has a unique limit.[4]

4.4 Properties

Subspaces and products of Hausdorff spaces are Hausdorff,[5] but quotient spaces of Hausdorff spaces need not beHausdorff. In fact, every topological space can be realized as the quotient of some Hausdorff space.[6]

Hausdorff spaces are T1, meaning that all singletons are closed. Similarly, preregular spaces are R0.Another nice property of Hausdorff spaces is that compact sets are always closed.[7] This may fail in non-Hausdorffspaces such as Sierpiński space.The definition of a Hausdorff space says that points can be separated by neighborhoods. It turns out that this impliessomething which is seemingly stronger: in a Hausdorff space every pair of disjoint compact sets can also be separatedby neighborhoods,[8] in other words there is a neighborhood of one set and a neighborhood of the other, such that thetwo neighborhoods are disjoint. This is an example of the general rule that compact sets often behave like points.Compactness conditions together with preregularity often imply stronger separation axioms. For example, any locallycompact preregular space is completely regular. Compact preregular spaces are normal, meaning that they satisfyUrysohn’s lemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite opencovers. The Hausdorff versions of these statements are: every locally compact Hausdorff space is Tychonoff, andevery compact Hausdorff space is normal Hausdorff.The following results are some technical properties regarding maps (continuous and otherwise) to and fromHausdorffspaces.

24 CHAPTER 4. HAUSDORFF SPACE

Let f : X → Y be a continuous function and suppose Y is Hausdorff. Then the graph of f, {(x, f(x)) | x ∈ X} , isa closed subset of X × Y.Let f : X → Y be a function and let ker(f) ≜ {(x, x′) | f(x) = f(x′)} be its kernel regarded as a subspace of X ×X.

• If f is continuous and Y is Hausdorff then ker(f) is closed.

• If f is an open surjection and ker(f) is closed then Y is Hausdorff.

• If f is a continuous, open surjection (i.e. an open quotient map) then Y is Hausdorff if and only if ker(f) isclosed.

If f,g : X→ Y are continuous maps and Y is Hausdorff then the equalizer eq(f, g) = {x | f(x) = g(x)} is closed inX. It follows that if Y is Hausdorff and f and g agree on a dense subset of X then f = g. In other words, continuousfunctions into Hausdorff spaces are determined by their values on dense subsets.Let f : X→ Y be a closed surjection such that f−1(y) is compact for all y ∈ Y. Then if X is Hausdorff so is Y.Let f : X→ Y be a quotient map with X a compact Hausdorff space. Then the following are equivalent

• Y is Hausdorff

• f is a closed map

• ker(f) is closed

4.5 Preregularity versus regularity

All regular spaces are preregular, as are all Hausdorff spaces. There are many results for topological spaces that holdfor both regular and Hausdorff spaces. Most of the time, these results hold for all preregular spaces; they were listedfor regular and Hausdorff spaces separately because the idea of preregular spaces came later. On the other hand,those results that are truly about regularity generally don't also apply to nonregular Hausdorff spaces.There are many situations where another condition of topological spaces (such as paracompactness or local com-pactness) will imply regularity if preregularity is satisfied. Such conditions often come in two versions: a regularversion and a Hausdorff version. Although Hausdorff spaces aren't generally regular, a Hausdorff space that is also(say) locally compact will be regular, because any Hausdorff space is preregular. Thus from a certain point of view,it is really preregularity, rather than regularity, that matters in these situations. However, definitions are usually stillphrased in terms of regularity, since this condition is better known than preregularity.See History of the separation axioms for more on this issue.

4.6 Variants

The terms “Hausdorff”, “separated”, and “preregular” can also be applied to such variants on topological spacesas uniform spaces, Cauchy spaces, and convergence spaces. The characteristic that unites the concept in all of theseexamples is that limits of nets and filters (when they exist) are unique (for separated spaces) or unique up to topologicalindistinguishability (for preregular spaces).As it turns out, uniform spaces, and more generally Cauchy spaces, are always preregular, so the Hausdorff condi-tion in these cases reduces to the T0 condition. These are also the spaces in which completeness makes sense, andHausdorffness is a natural companion to completeness in these cases. Specifically, a space is complete if and only ifevery Cauchy net has at least one limit, while a space is Hausdorff if and only if every Cauchy net has at most onelimit (since only Cauchy nets can have limits in the first place).

4.7. ALGEBRA OF FUNCTIONS 25

4.7 Algebra of functions

The algebra of continuous (real or complex) functions on a compact Hausdorff space is a commutative C*-algebra,and conversely by the Banach–Stone theorem one can recover the topology of the space from the algebraic propertiesof its algebra of continuous functions. This leads to noncommutative geometry, where one considers noncommutativeC*-algebras as representing algebras of functions on a noncommutative space.

4.8 Academic humour• Hausdorff condition is illustrated by the pun that in Hausdorff spaces any two points can be “housed off” fromeach other by open sets.[9]

• In the Mathematics Institute of at the University of Bonn, in which Felix Hausdorff researched and lectured,there is a certain room designated the Hausdorff-Raum. This is a pun, as Raum means both room and spacein German.

4.9 See also• Quasitopological space

• Weak Hausdorff space

• Fixed-point space, a Hausdorff space X such that every continuous function f:X→X has a fixed point.

4.10 Notes[1] Willard, pp. 86–87.

[2] Willard, pp. 86–87.

[3] Bourbaki, p. 75.

[4] van Douwen, Eric K. (1993). “An anti-Hausdorff Fréchet space in which convergent sequences have unique limits”.Topology and its Applications 51 (2): 147–158. doi:10.1016/0166-8641(93)90147-6.

[5] Hausdorff property is hereditary at PlanetMath.org.

[6] Shimrat, M. (1956). “Decomposition spaces and separation properties”. Quart. J. Math. 2: 128–129.

[7] Proof of A compact set in a Hausdorff space is closed at PlanetMath.org.

[8] Willard, p. 124.

[9] Colin Adams and Robert Franzosa. Introduction to Topology: Pure and Applied. p. 42

4.11 References• Arkhangelskii, A.V., L.S. Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4.

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

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

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

Chapter 5

Neighbourhood (mathematics)

For the concept in graph theory, see Neighbourhood (graph theory).In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts

A set V in the plane is a neighbourhood of a point p if a small disk around p is contained in V .

in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where one canmove that point some amount without leaving the set.

26

5.1. DEFINITION 27

pV

A rectangle is not a neighbourhood of any of its corners.

This concept is closely related to the concepts of open set and interior.

5.1 Definition

If X is a topological space and p is a point in X , a neighbourhood of p is a subset V of X that includes an openset U containing p ,

p ∈ U ⊆ V.

This is also equivalent to p ∈ X being in the interior of V .Note that the neighbourhood V need not be an open set itself. If V is open it is called an open neighbourhood.Some scholars require that neighbourhoods be open, so it is important to note conventions.A set that is a neighbourhood of each of its points is open since it can be expressed as the union of open sets containingeach of its points.The collection of all neighbourhoods of a point is called the neighbourhood system at the point.If S is a subset ofX then a neighbourhood of S is a set V that includes an open set U containing S . It follows thata set V is a neighbourhood of S if and only if it is a neighbourhood of all the points in S . Furthermore, it followsthat V is a neighbourhood of S iff S is a subset of the interior of V .

5.2 In a metric space

In a metric space M = (X, d) , a set V is a neighbourhood of a point p if there exists an open ball with centre pand radius r > 0 , such that

28 CHAPTER 5. NEIGHBOURHOOD (MATHEMATICS)

A set S in the plane and a uniform neighbourhood V of S .

aa-ε a+ε

The epsilon neighbourhood of a number a on the real number line.

Br(p) = B(p; r) = {x ∈ X | d(x, p) < r}

is contained in V .V is called uniform neighbourhood of a set S if there exists a positive number r such that for all elements p of S ,

Br(p) = {x ∈ X | d(x, p) < r}

is contained in V .For r > 0 the r -neighbourhood Sr of a set S is the set of all points in X that are at distance less than r from S(or equivalently, S r is the union of all the open balls of radius r that are centred at a point in S ).It directly follows that an r -neighbourhood is a uniform neighbourhood, and that a set is a uniform neighbourhoodif and only if it contains an r -neighbourhood for some value of r .

5.3. EXAMPLES 29

M

( )a-ε a+εa

( [ )) [ ]

The set M is a neighbourhood of the number a, because there is an ε-neighbourhood of a which is a subset of M.

5.3 Examples

Given the set of real numbers R with the usual Euclidean metric and a subset V defined as

V :=∪n∈N

B (n ; 1/n) ,

then V is a neighbourhood for the set N of natural numbers, but is not a uniform neighbourhood of this set.

5.4 Topology from neighbourhoods

The above definition is useful if the notion of open set is already defined. There is an alternative way to define atopology, by first defining the neighbourhood system, and then open sets as those sets containing a neighbourhood ofeach of their points.A neighbourhood system onX is the assignment of a filter N(x) (on the set X ) to each x in X , such that

1. the point x is an element of each U in N(x)

2. each U in N(x) contains some V in N(x) such that for each y in V , U is in N(y) .

One can show that both definitions are compatible, i.e. the topology obtained from the neighbourhood system definedusing open sets is the original one, and vice versa when starting out from a neighbourhood system.

5.5 Uniform neighbourhoods

In a uniform space S = (X, δ) , V is called a uniform neighbourhood of P if P is not close toX \V , that is thereexists no entourage containing P and X \ V .

5.6 Deleted neighbourhood

A Deleted neighbourhood of a point p (sometimes called a punctured neighbourhood) is a neighbourhood of p ,without {p} . For instance, the interval (−1, 1) = {y : −1 < y < 1} is a neighbourhood of p = 0 in the real line,so the set (−1, 0)∪ (0, 1) = (−1, 1) \ {0} is a deleted neighbourhood of 0 . Note that a deleted neighbourhood of agiven point is not in fact a neighbourhood of the point. The concept of deleted neighbourhood occurs in the definitionof the limit of a function.

5.7 See also

• Tubular neighborhood

30 CHAPTER 5. NEIGHBOURHOOD (MATHEMATICS)

5.8 References• Kelley, John L. (1975). General topology. New York: Springer-Verlag. ISBN 0-387-90125-6.

• Bredon, Glen E. (1993). Topology and geometry. New York: Springer-Verlag. ISBN 0-387-97926-3.

• Kaplansky, Irving (2001). Set Theory and Metric Spaces. American Mathematical Society. ISBN 0-8218-2694-8.

Chapter 6

Open set

Example: The points (x, y) satisfying x2 + y2 = r2 are colored blue. The points (x, y) satisfying x2 + y2 < r2 are colored red. Thered points form an open set. The blue points form a boundary set. The union of the red and blue points is a closed set.

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line. The simplestexample is in metric spaces, where open sets can be defined as those sets which contain an open ball around each of

31

32 CHAPTER 6. OPEN SET

their points (or, equivalently, a set is open if it doesn't contain any of its boundary points); however, an open set, ingeneral, can be very abstract: any collection of sets can be called open, as long as the union of an arbitrary numberof open sets is open, the intersection of a finite number of open sets is open, and the space itself is open. Theseconditions are very loose, and they allow enormous flexibility in the choice of open sets. In the two extremes, everyset can be open (called the discrete topology), or no set can be open but the space itself (the indiscrete topology).In practice, however, open sets are usually chosen to be similar to the open intervals of the real line. The notion of anopen set provides a fundamental way to speak of nearness of points in a topological space, without explicitly havinga concept of distance defined. Once a choice of open sets is made, the properties of continuity, connectedness, andcompactness, which use notions of nearness, can be defined using these open sets.Each choice of open sets for a space is called a topology. Although open sets and the topologies that they comprise areof central importance in point-set topology, they are also used as an organizational tool in other important branchesof mathematics. Examples of topologies include the Zariski topology in algebraic geometry that reflects the algebraicnature of varieties, and the topology on a differential manifold in differential topology where each point within thespace is contained in an open set that is homeomorphic to an open ball in a finite-dimensional Euclidean space.

6.1 Motivation

Intuitively, an open set provides a method to distinguish two points. For example, if about one point in a topologicalspace there exists an open set not containing another (distinct) point, the two points are referred to as topologicallydistinguishable. In this manner, one may speak of whether two subsets of a topological space are “near” withoutconcretely defining a metric on the topological space. Therefore, topological spaces may be seen as a generalizationof metric spaces.In the set of all real numbers, one has the natural Euclidean metric; that is, a function which measures the distancebetween two real numbers: d(x, y) = |x - y|. Therefore, given a real number, one can speak of the set of all pointsclose to that real number; that is, within ε of that real number (refer to this real number as x). In essence, pointswithin ε of x approximate x to an accuracy of degree ε. Note that ε > 0 always but as ε becomes smaller and smaller,one obtains points that approximate x to a higher and higher degree of accuracy. For example, if x = 0 and ε = 1, thepoints within ε of x are precisely the points of the interval (−1, 1); that is, the set of all real numbers between −1and 1. However, with ε = 0.5, the points within ε of x are precisely the points of (−0.5, 0.5). Clearly, these pointsapproximate x to a greater degree of accuracy compared to when ε = 1.The previous discussion shows, for the case x = 0, that one may approximate x to higher and higher degrees ofaccuracy by defining ε to be smaller and smaller. In particular, sets of the form (-ε, ε) give us a lot of informationabout points close to x = 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describepoints close to x. This innovative idea has far-reaching consequences; in particular, by defining different collectionsof sets containing 0 (distinct from the sets (-ε, ε)), one may find different results regarding the distance between 0 andother real numbers. For example, if we were to define R as the only such set for “measuring distance”, all points areclose to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a memberof R. Thus, we find that in some sense, every real number is distance 0 away from 0! It may help in this case to thinkof the measure as being a binary condition, all things in R are equally close to 0, while any item that is not in R is notclose to 0.In general, one refers to the family of sets containing 0, used to approximate 0, as a neighborhood basis; a memberof this neighborhood basis is referred to as an open set. In fact, one may generalize these notions to an arbitraryset (X); rather than just the real numbers. In this case, given a point (x) of that set, one may define a collectionof sets “around” (that is, containing) x, used to approximate x. Of course, this collection would have to satisfycertain properties (known as axioms) for otherwise we may not have a well-defined method to measure distance. Forexample, every point in X should approximate x to some degree of accuracy. Thus X should be in this family. Oncewe begin to define “smaller” sets containing x, we tend to approximate x to a greater degree of accuracy. Bearing thisin mind, one may define the remaining axioms that the family of sets about x is required to satisfy.

6.2 Definitions

The concept of open sets can be formalized with various degrees of generality, for example:

6.3. PROPERTIES 33

6.2.1 Euclidean space

A subset U of the Euclidean n-space Rn is called open if, given any point x in U, there exists a real number ε > 0 suchthat, given any point y in Rn whose Euclidean distance from x is smaller than ε, y also belongs to U.[1] Equivalently,a subset U of Rn is open if every point in U has a neighborhood in Rn contained in U.

6.2.2 Metric spaces

A subset U of a metric space (M, d) is called open if, given any point x in U, there exists a real number ε > 0 suchthat, given any point y in M with d(x, y) < ε, y also belongs to U. Equivalently, U is open if every point in U has aneighbourhood contained in U.This generalises the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.

6.2.3 Topological spaces

In general topological spaces, the open sets can be almost anything, with different choices giving different spaces.Let X be a set and τ be a family of sets. We say that τ is a topology on X if:

• X ∈ τ, ∅ ∈ τ ( X and ∅ are in τ )

• {Oi}i∈I ⊆ τ ⇒ ∪i∈IOi ∈ τ (any union of sets in τ is in τ )

• {Oi}i∈I ⊆ τ ⇒ ∩ni=1Oi ∈ τ (any finite intersection of sets in τ is in τ )

We call the sets in τ the open sets.Note that infinite intersections of open sets need not be open. For example, the intersection of all intervals of theform (−1/n, 1/n), where n is a positive integer, is the set {0} which is not open in the real line. Sets that can beconstructed as the intersection of countably many open sets are denoted Gδ sets.The topological definition of open sets generalises the metric space definition: If one begins with a metric space anddefines open sets as before, then the family of all open sets is a topology on the metric space. Every metric space istherefore, in a natural way, a topological space. There are, however, topological spaces that are not metric spaces.

6.3 Properties

• The empty set is both open and closed (clopen set).[2]

• The set X that the topology is defined on is both open and closed (clopen set).

• The union of any number of open sets is open.[3]

• The intersection of a finite number of open sets is open.[3]

6.4 Uses

Open sets have a fundamental importance in topology. The concept is required to define andmake sense of topologicalspace and other topological structures that deal with the notions of closeness and convergence for spaces such as metricspaces and uniform spaces.Every subset A of a topological space X contains a (possibly empty) open set; the largest such open set is called theinterior of A. It can be constructed by taking the union of all the open sets contained in A.Given topological spaces X and Y, a function f from X to Y is continuous if the preimage of every open set in Y isopen in X. The function f is called open if the image of every open set in X is open in Y.An open set on the real line has the characteristic property that it is a countable union of disjoint open intervals.

34 CHAPTER 6. OPEN SET

6.5 Notes and cautions

6.5.1 “Open” is defined relative to a particular topology

Whether a set is open depends on the topology under consideration. Having opted for greater brevity over greaterclarity, we refer to a set X endowed with a topology T as “the topological space X" rather than “the topological space(X, T)", despite the fact that all the topological data is contained in T. If there are two topologies on the same set, a setU that is open in the first topology might fail to be open in the second topology. For example, if X is any topologicalspace and Y is any subset of X, the set Y can be given its own topology (called the 'subspace topology') defined by“a set U is open in the subspace topology on Y if and only if U is the intersection of Y with an open set from theoriginal topology on X.” This potentially introduces new open sets: if V is open in the original topology on X, butV ∩ Y isn't, then V ∩ Y is open in the subspace topology on Y but not in the original topology on X.As a concrete example of this, if U is defined as the set of rational numbers in the interval (0, 1), then U is an opensubset of the rational numbers, but not of the real numbers. This is because when the surrounding space is the rationalnumbers, for every point x in U, there exists a positive number a such that all rational points within distance a ofx are also in U. On the other hand, when the surrounding space is the reals, then for every point x in U there is nopositive a such that all real points within distance a of x are in U (since U contains no non-rational numbers).

6.5.2 Open and closed are not mutually exclusive

A set might be open, closed, both, or neither.For example, we'll use the real line with its usual topology (the Euclidean topology), which is defined as follows:every interval (a,b) of real numbers belongs to the topology, and every union of such intervals, e.g. (a, b) ∪ (c, d) ,belongs to the topology.

• In any topology, the entire set X is declared open by definition, as is the empty set. Moreover, the complementof the entire set X is the empty set; since X has an open complement, this means by definition that X is closed.Hence, in any topology, the entire space is simultaneously open and closed ("clopen").

• The interval I = (0, 1) is open because it belongs to the Euclidean topology. If I were to have an opencomplement, it would mean by definition that I were closed. But I does not have an open complement; itscomplement is IC = (−∞, 0] ∪ [1,∞) , which does not belong to the Euclidean topology since it is not aunion of intervals of the form (a, b) . Hence, I is an example of a set that is open but not closed.

• By a similar argument, the interval J = [0, 1] is closed but not open.• Finally, since neither K = [0, 1) nor its complement KC = (−∞, 0) ∪ [1,∞) belongs to the Euclideantopology (neither one can be written as a union of intervals of the form (a,b) ), this means that K is neitheropen nor closed.

6.6 See also• Closed set• Clopen set• Neighbourhood

6.7 References[1] Ueno, Kenji et al. (2005). “The birth of manifolds”. A Mathematical Gift: The Interplay Between Topology, Functions,

Geometry, and Algebra. Vol. 3. American Mathematical Society. p. 38. ISBN 9780821832844.

[2] Krantz, Steven G. (2009). “Fundamentals”. Essentials of Topology With Applications. CRC Press. pp. 3–4. ISBN9781420089745.

[3] Taylor, Joseph L. (2011). “Analytic functions”. Complex Variables. The Sally Series. American Mathematical Society. p.29. ISBN 9780821869017.

6.8. EXTERNAL LINKS 35

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

• Open Set at PlanetMath.org.

Chapter 7

Separable space

Not to be confused with Separated space.

In mathematics a topological space is called separable if it contains a countable, dense subset; that is, there exists asequence {xn}∞n=1 of elements of the space such that every nonempty open subset of the space contains at least oneelement of the sequence.Like the other axioms of countability, separability is a “limitation on size”, not necessarily in terms of cardinality(though, in the presence of the Hausdorff axiom, this does turn out to be the case; see below) but in a more subtletopological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorffspace is determined by its values on the countable dense subset.Contrast separability with the related notion of second countability, which is in general stronger but equivalent on theclass of metrizable spaces.

7.1 First examples

Any topological space which is itself finite or countably infinite is separable, for the whole space is a countable densesubset of itself. An important example of an uncountable separable space is the real line, in which the rationalnumbers form a countable dense subset. Similarly the set of all vectors (r1, . . . , rn) ∈ Rn in which ri is rational forall i is a countable dense subset of Rn ; so for every n the n -dimensional Euclidean space is separable.A simple example of a space which is not separable is a discrete space of uncountable cardinality.Further examples are given below.

7.2 Separability versus second countability

Any second-countable space is separable: if {Un} is a countable base, choosing any xn∈Un from the non-empty Un

gives a countable dense subset. Conversely, a metrizable space is separable if and only if it is second countable, whichis the case if and only if it is Lindelöf.To further compare these two properties:

• An arbitrary subspace of a second countable space is second countable; subspaces of separable spaces need notbe separable (see below).

• Any continuous image of a separable space is separable (Willard 1970, Th. 16.4a).; even a quotient of a secondcountable space need not be second countable.

• A product of at most countably many separable spaces is separable. A countable product of second countablespaces is second countable, but an uncountable product of second countable spaces need not even be firstcountable.

36

7.3. CARDINALITY 37

7.3 Cardinality

The property of separability does not in and of itself give any limitations on the cardinality of a topological space: anyset endowed with the trivial topology is separable, as well as second countable, quasi-compact, and connected. The“trouble” with the trivial topology is its poor separation properties: its Kolmogorov quotient is the one-point space.A first countable, separable Hausdorff space (in particular, a separable metric space) has at most the continuumcardinality c . In such a space, closure is determined by limits of sequences and any convergent sequence has at mostone limit, so there is a surjective map from the set of convergent sequences with values in the countable dense subsetto the points of X .A separable Hausdorff space has cardinality at most 2c , where c is the cardinality of the continuum. For this closureis characterized in terms of limits of filter bases: if Y ⊆ X and z ∈ X , then z ∈ Y if and only if there exists afilter base B consisting of subsets of Y which converges to z . The cardinality of the set S(Y ) of such filter bases isat most 22|Y | . Moreover, in a Hausdorff space, there is at most one limit to every filter base. Therefore, there is asurjection S(Y ) → X when Y = X.

The same arguments establish a more general result: suppose that a Hausdorff topological space X contains a densesubset of cardinality κ . Then X has cardinality at most 22κ and cardinality at most 2κ if it is first countable.The product of at most continuum many separable spaces is a separable space (Willard 1970, p. 109, Th 16.4c). Inparticular the space RR of all functions from the real line to itself, endowed with the product topology, is a separableHausdorff space of cardinality 2c . More generally, if κ is any infinite cardinal, then a product of at most 2κ spaceswith dense subsets of size at most κ has itself a dense subset of size at most κ (Hewitt–Marczewski–Pondiczerytheorem).

7.4 Constructive mathematics

Separability is especially important in numerical analysis and constructive mathematics, since many theorems that canbe proved for nonseparable spaces have constructive proofs only for separable spaces. Such constructive proofs canbe turned into algorithms for use in numerical analysis, and they are the only sorts of proofs acceptable in constructiveanalysis. A famous example of a theorem of this sort is the Hahn–Banach theorem.

7.5 Further examples

7.5.1 Separable spaces

• Every compact metric space (or metrizable space) is separable.

• The space C(K) of all continuous functions from a compact subsetK ⊆ R to the real line R is separable.

• The Lebesgue spacesLp (X,µ) , over a separable measure space ⟨X,M, µ⟩ , are separable for any 1 ≤ p < ∞.

• Any topological space which is the union of a countable number of separable subspaces is separable. Together,these first two examples give a different proof that n -dimensional Euclidean space is separable.

• The space C([0, 1]) of continuous real-valued functions on the unit interval [0, 1] with the metric of uniformconvergence is a separable space, since it follows from the Weierstrass approximation theorem that the setQ[x] of polynomials in one variable with rational coefficients is a countable dense subset of C([0, 1]) . TheBanach-Mazur theorem asserts that any separable Banach space is isometrically isomorphic to a closed linearsubspace of C([0, 1]) .

• A Hilbert space is separable if and only if it has a countable orthonormal basis. It follows that any separable,infinite-dimensional Hilbert space is isometric to the space ℓ2 of square-summable sequences.

• An example of a separable space that is not second-countable is the Sorgenfrey line S , the set of real numbersequipped with the lower limit topology.

38 CHAPTER 7. SEPARABLE SPACE

7.5.2 Non-separable spaces

• The first uncountable ordinal ω1 , equipped with its natural order topology, is not separable.

• The Banach space ℓ∞ of all bounded real sequences, with the supremum norm, is not separable. The sameholds for L∞ .

• The Banach space of functions of bounded variation is not separable; note however that this space has veryimportant applications in mathematics, physics and engineering.

7.6 Properties• A subspace of a separable space need not be separable (see the Sorgenfrey plane and the Moore plane), butevery open subspace of a separable space is separable, (Willard 1970, Th 16.4b). Also every subspace of aseparable metric space is separable.

• In fact, every topological space is a subspace of a separable space of the same cardinality. A constructionadding at most countably many points is given in (Sierpinski 1952, p. 49); if the space was a Hausdorff spacethen the space constructed which it embeds into is also a Hausdorff space.

• The set of all real-valued continuous functions on a separable space has a cardinality less than or equal to c.This follows since such functions are determined by their values on dense subsets.

• From the above property, one can deduce the following: If X is a separable space having an uncountable closeddiscrete subspace, then X cannot be normal. This shows that the Sorgenfrey plane is not normal.

• For a compact Hausdorff space X, the following are equivalent:

(i) X is second countable.(ii) The space C(X,R) of continuous real-valued functions on X with the supremum norm isseparable.(iii) X is metrizable.

7.6.1 Embedding separable metric spaces

• Every separable metric space is homeomorphic to a subset of the Hilbert cube. This is established in the proofof the Urysohn metrization theorem.

• Every separable metric space is isometric to a subset of the (non-separable) Banach space l∞ of all boundedreal sequences with the supremum norm; this is known as the Fréchet embedding. (Heinonen 2003)

• Every separable metric space is isometric to a subset of C([0,1]), the separable Banach space of continuousfunctions [0,1]→R, with the supremum norm. This is due to Stefan Banach. (Heinonen 2003)

• Every separable metric space is isometric to a subset of the Urysohn universal space.

For nonseparable spaces:

• A metric space of density equal to an infinite cardinal α is isometric to a subspace of C([0,1]α, R), the spaceof real continuous functions on the product of α copies of the unit interval. (Kleiber 1969)

7.7 References• Heinonen, Juha (January 2003), Geometric embeddings of metric spaces (PDF), retrieved 6 February 2009

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

7.7. REFERENCES 39

• Kleiber, Martin; Pervin, William J. (1969), “A generalized Banach-Mazur theorem”, Bull. Austral. Math. Soc.1: 169–173, doi:10.1017/S0004972700041411

• Sierpiński, Wacław (1952), General topology, Mathematical Expositions, No. 7, Toronto, Ont.: University ofToronto Press, MR 0050870

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

• Willard, Stephen (1970), General Topology, Addison-Wesley, ISBN 978-0-201-08707-9, MR 0264581

Chapter 8

Subset

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

AB

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

40

8.1. DEFINITIONS 41

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

8.1 Definitions

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

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

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

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

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

or equivalently

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

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

8.2 ⊂ and ⊃ symbols

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

8.3 Examples

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

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

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

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

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

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

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

42 CHAPTER 8. SUBSET

polygonsregular

polygons

The regular polygons form a subset of the polygons

Another example in an Euler diagram:

• A is a proper subset of B

• C is a subset but not a proper subset of B

8.4 Other properties of inclusion

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

8.5 See also• Containment order

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

978-0-07-338309-5.

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

8.7. EXTERNAL LINKS 43

C B A

A B and B C imply A C

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

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

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

Chapter 9

Topological space

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

9.1 History of Development

9.2 Definition

Main article: Characterizations of the category of topological spaces

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

9.2.1 Neighbourhoods definition

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

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

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

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

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

44

9.2. DEFINITION 45

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

9.2.2 Open sets definition

1 2 3 1 2 3

1 2 3 1 2 3

1 2 3 1 2 3

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

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

1. The empty set and X itself are open.

2. Any union of open sets is open.

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

46 CHAPTER 9. TOPOLOGICAL SPACE

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

Examples

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

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

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

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

9.2.3 Closed sets definition

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

1. The empty set and X are closed.

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

3. The union of any finite number of closed sets is also closed.

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

9.2.4 Other definitions

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

9.3 Comparison of topologies

Main article: Comparison of topologies

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

9.4. CONTINUOUS FUNCTIONS 47

9.4 Continuous functions

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

9.5 Examples of topological spaces

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

48 CHAPTER 9. TOPOLOGICAL SPACE

many distinct topologies defined on it.If Γ is an ordinal number, then the set Γ = [0, Γ) may be endowed with the order topology generated by the intervals(a, b), [0, b) and (a, Γ) where a and b are elements of Γ.

9.6 Topological constructions

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

9.7 Classification of topological spaces

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

9.8 Topological spaces with algebraic structure

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

9.9 Topological spaces with order structure• Spectral. A space is spectral if and only if it is the prime spectrum of a ring (Hochster theorem).

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

9.10 Specializations and generalizations

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

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

9.11. SEE ALSO 49

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

• Uniform spaces axiomatize ordering the distance between distinct points.

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

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

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

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

9.11 See also• Space (mathematics)

• Kolmogorov space (T0)

• accessible/Fréchet space (T1)

• Hausdorff space (T2)

• Completely Hausdorff space and Urysohn space (T₂½)

• Regular space and regular Hausdorff space (T3)

• Tychonoff space and completely regular space (T₃½)

• Normal Hausdorff space (T4)

• Completely normal Hausdorff space (T5)

• Perfectly normal Hausdorff space (T6)

• Quasitopological space

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

9.12 Notes[1] Schubert 1968, p. 13

[2] Brown 2006, section 2.1.

[3] Brown 2006, section 2.2.

[4] Armstrong 1983, definition 2.1.

[5] Armstrong 1983, theorem 2.6.

9.13 References• Armstrong, M. A. (1983) [1979]. Basic Topology. Undergraduate Texts in Mathematics. Springer. ISBN0-387-90839-0.

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

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

50 CHAPTER 9. TOPOLOGICAL SPACE

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

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

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

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

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

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

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

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

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

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

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

• Topological space at PlanetMath.org.

Chapter 10

Topology

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

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

sets, making a given set a topological space. It is an area of mathematics concerned with the properties of space thatare preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. Importanttopological properties include connectedness and compactness.[1]

Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space,dimension, and transformation.[2] Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned thegeometria situs (Greek-Latin for “geometry of place”) and analysis situs (Greek-Latin for “picking apart of place”).Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field’s first theo-rems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not untilthe first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20thcentury, topology had become a major branch of mathematics.Topology has many subfields:

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

51

52 CHAPTER 10. TOPOLOGY

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

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

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

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

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

10.1 History

Topology began with the investigation of certain questions in geometry. Leonhard Euler's 1736 paper on the SevenBridges of Königsberg[3] is regarded as one of the first academic treatises in modern topology.The term “Topologie” was introduced in German in 1847 by Johann Benedict Listing in Vorstudien zur Topologie,[4]who had used the word for ten years in correspondence before its first appearance in print. The English form topologywas first used in 1883 in Listing’s obituary in the journal Nature[5] to distinguish "...qualitative geometry from theordinary geometry in which quantitative relations chiefly are treated.” The term topologist in the sense of a specialistin topology was used in 1905 in the magazine Spectator. However, none of these uses corresponds exactly to themodern definition of topology.Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19thcentury. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space aspart of his study of Fourier series.

10.2. INTRODUCTION 53

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

Henri Poincaré published Analysis Situs in 1895,[6] introducing the concepts of homotopy and homology, which arenow considered part of algebraic topology.Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, GiulioAscoli and others, Maurice Fréchet introduced the metric space in 1906.[7] A metric space is now considered aspecial case of a general topological space. In 1914, Felix Hausdorff coined the term “topological space” and gavethe definition for what is now called a Hausdorff space.[8] Currently, a topological space is a slight generalization ofHausdorff spaces, given in 1922 by Kazimierz Kuratowski.For further developments, see point-set topology and algebraic topology.

10.2 Introduction

Topology can be formally defined as “the study of qualitative properties of certain objects (called topological spaces)that are invariant under a certain kind of transformation (called a continuous map), especially those properties thatare invariant under a certain kind of transformation (called homeomorphism).”Topology is also used to refer to a structure imposed upon a set X, a structure that essentially 'characterizes’ the set Xas a topological space by taking proper care of properties such as convergence, connectedness and continuity, upontransformation.Topological spaces show up naturally in almost every branch of mathematics. This has made topology one of thegreat unifying ideas of mathematics.The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objectsinvolved, but rather on the way they are put together. For example, the square and the circle have many properties incommon: they are both one dimensional objects (from a topological point of view) and both separate the plane intotwo parts, the part inside and the part outside.

54 CHAPTER 10. TOPOLOGY

In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route throughthe town of Königsberg (now Kaliningrad) that would cross each of its seven bridges exactly once. This result didnot depend on the lengths of the bridges, nor on their distance from one another, but only on connectivity properties:which bridges connect to which islands or riverbanks. This problem in introductory mathematics called Seven Bridgesof Königsberg led to the branch of mathematics known as graph theory.

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

Similarly, the hairy ball theorem of algebraic topology says that “one cannot comb the hair flat on a hairy ball withoutcreating a cowlick.” This fact is immediately convincing to most people, even though they might not recognize themore formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere.As with the Bridges of Königsberg, the result does not depend on the shape of the sphere; it applies to any kind ofsmooth blob, as long as it has no holes.To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just whatproperties these problems do rely on. From this need arises the notion of homeomorphism. The impossibility ofcrossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and thehairy ball theorem applies to any space homeomorphic to a sphere.Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditionaljoke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut couldbe reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.

10.3. CONCEPTS 55

Homeomorphism can be considered the most basic topological equivalence. Another is homotopy equivalence. Thisis harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent ifthey both result from “squishing” some larger object.An introductory exercise is to classify the uppercase letters of the English alphabet according to homeomorphismand homotopy equivalence. The result depends partially on the font used. The figures use the sans-serif Myriad font.Homotopy equivalence is a rougher relationship than homeomorphism; a homotopy equivalence class can containseveral homeomorphism classes. The simple case of homotopy equivalence described above can be used here toshow two letters are homotopy equivalent. For example, O fits inside P and the tail of the P can be squished to the“hole” part.Homeomorphism classes are:

• no holes,

• no holes three tails,

• no holes four tails,

• one hole no tail,

• one hole one tail,

• one hole two tails,

• two holes no tail, and

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

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

• one hole,

• two holes, and

• no holes.

To classify the letters correctly, we must show that two letters in the same class are equivalent and two letters indifferent classes are not equivalent. In the case of homeomorphism, this can be done by selecting points and showingtheir removal disconnects the letters differently. For example, X and Y are not homeomorphic because removingthe center point of the X leaves four pieces; whatever point in Y corresponds to this point, its removal can leave atmost three pieces. The case of homotopy equivalence is harder and requires a more elaborate argument showing analgebraic invariant, such as the fundamental group, is different on the supposedly differing classes.Letter topology has practical relevance in stencil typography. For instance, Braggadocio font stencils are made of oneconnected piece of material.

10.3 Concepts

10.3.1 Topologies on Sets

Main article: Topological space

The term topology also refers to a specific mathematical idea central to the area of mathematics called topology.Informally, a topology tells how elements of a set relate spatially to each other. The same set can have differenttopologies. For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set withdifferent topologies.Formally, let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:

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

56 CHAPTER 10. TOPOLOGY

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

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

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

10.3.2 Continuous functions and homeomorphisms

Main articles: Continuous function and homeomorphism

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

10.3.3 Manifolds

Main article: Manifold

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

10.4 Topics

10.4.1 General topology

Main article: General topology

General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions usedin topology.[9][10] It is the foundation of most other branches of topology, including differential topology, geometrictopology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness. Intuitively, continu-ous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many setsof arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The wordsnearby, arbitrarily small, and far apart can all be made precise by using open sets. If we change the definition ofopen set, we change what continuous functions, compact sets, and connected sets are. Each choice of definition foropen set is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric.Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.

10.4. TOPICS 57

10.4.2 Algebraic topology

Main article: Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.[11]The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usuallymost classify up to homotopy equivalence.The most important of these invariants are homotopy groups, homology, and cohomology.Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraicproblems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroupof a free group is again a free group.

10.4.3 Differential topology

Main article: Differential topology

Differential topology is the field dealing with differentiable functions on differentiable manifolds.[12] It is closelyrelated to differential geometry and together they make up the geometric theory of differentiable manifolds.More specifically, differential topology considers the properties and structures that require only a smooth structureon a manifold to be defined. Smooth manifolds are 'softer' than manifolds with extra geometric structures, whichcan act as obstructions to certain types of equivalences and deformations that exist in differential topology. Forinstance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on thesame smooth manifold—that is, one can smoothly “flatten out” certain manifolds, but it might require distorting thespace and affecting the curvature or volume.

10.4.4 Geometric topology

Main article: Geometric topology

Geometric topology is a branch of topology that primarily focuses on low-dimensional manifolds (i.e. dimensions2,3 and 4) and their interaction with geometry, but it also includes some higher-dimensional topology.[13] [14] Someexamples of topics in geometric topology are orientability, handle decompositions, local flatness, and the planar andhigher-dimensional Schönflies theorem.In high-dimensional topology, characteristic classes are a basic invariant, and surgery theory is a key theory.Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – ev-ery surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curva-ture/spherical, zero curvature/flat, negative curvature/hyperbolic – and the geometrization conjecture (now theorem)in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of eight possible geometries.2-dimensional topology can be studied as complex geometry in one variable (Riemann surfaces are complex curves)– by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces),though not every 4-manifold admits a complex structure.

10.4.5 Generalizations

Occasionally, one needs to use the tools of topology but a “set of points” is not available. In pointless topology oneconsiders instead the lattice of open sets as the basic notion of the theory,[15] while Grothendieck topologies arestructures defined on arbitrary categories that allow the definition of sheaves on those categories, and with that thedefinition of general cohomology theories.[16]

58 CHAPTER 10. TOPOLOGY

10.5 Applications

10.5.1 Biology

Knot theory, a branch of topology, is used in biology to study the effects of certain enzymes on DNA. These enzymescut, twist, and reconnect the DNA, causing knotting with observable effects such as slower electrophoresis.[17] Topol-ogy is also used in evolutionary biology to represent the relationship between phenotype and genotype.[18] Phenotypicforms that appear quite different can be separated by only a few mutations depending on how genetic changes mapto phenotypic changes during development.

10.5.2 Computer science

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

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

2. Analyse these topological complexes via algebraic topology — specifically, via the theory of persistent homol-ogy.[19]

3. Encode the persistent homology of a data set in the form of a parameterized version of a Betti number, whichis called a barcode.[19]

10.5.3 Physics

In physics, topology is used in several areas such as quantum field theory and cosmology.A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computestopological invariants.Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among otherthings, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces inalgebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topo-logical field theory.In cosmology, topology can be used to describe the overall shape of the universe.[20] This area is known as spacetimetopology.

10.5.4 Robotics

The various possible positions of a robot can be described by a manifold called configuration space.[21] In the area ofmotion planning, one finds paths between two points in configuration space. These paths represent a motion of therobot’s joints and other parts into the desired location and pose.

10.6 See also• Equivariant topology

• General topology

• List of algebraic topology topics

• List of examples in general topology

• List of general topology topics

• List of geometric topology topics

10.7. REFERENCES 59

• List of topology topics

• Publications in topology

• Topology glossary

• Topological space

10.7 References

[1] http://dictionary.reference.com/browse/topology

[2] http://www.math.wayne.edu/~{}rrb/topology.html

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

60 CHAPTER 10. TOPOLOGY

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

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

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

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

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

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

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

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

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

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

• Topology at DMOZ

• The Topological Zoo at The Geometry Center.

• Topology Atlas

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

• Topology Glossary

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

10.10. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 61

10.10 Text and image sources, contributors, and licenses

10.10.1 Text• Closed set Source: https://en.wikipedia.org/wiki/Closed_set?oldid=681209723 Contributors: AxelBoldt, Andre Engels, Toby~enwiki,

Toby Bartels, Someone else, Patrick, Smelialichu, Michael Hardy, Isomorphic, Dineshjk, Salsa Shark, AugPi, Dcoetzee, Zoicon5, Rob-bot, Tobias Bergemann, Tosha, Giftlite, Ævar Arnfjörð Bjarmason, Lethe, Fropuff, Python eggs, Karl-Henner, PhotoBox, Eep², PaulAugust, Themusicgod1, Cretog8, O18, JohnyDog, Msh210, ABCD, Sligocki, Oleg Alexandrov, Jannex, Isnow, Justin Ormont, MagisterMathematicae, BD2412, Salix alba, Marozols, VKokielov, Chobot, Buggi22, Roboto de Ajvol, RussBot, Grubber, Poulpy, Sardanaphalus,KnightRider~enwiki, SmackBot, Maksim-e~enwiki, JAnDudík, AGeek Tragedy, Dreadstar, Jim.belk, Mets501, J Di, Jackzhp, Jsd, Mya-suda, Thijs!bot, Colin Rowat, Salgueiro~enwiki, Rycee, Ddxc, DragonBot, Josephvk, Me314, Addbot, SomeUsr, Zorrobot, KamikazeBot,Ciphers, NickK, Xqbot, Bdmy, Almabot, Kamixave, RibotBOT, Erik9bot, Stpasha, EmausBot, Irina Gelbukh, DennisIsMe, ClueBot NG,Wcherowi, Langing, Paolo Lipparini, Charismaa, Marius siuram and Anonymous: 38

• Empty set Source: https://en.wikipedia.org/wiki/Empty_set?oldid=679013611 Contributors: AxelBoldt, Lee Daniel Crocker, Uriyan,Bryan Derksen, Tarquin, Jeronimo, Andre Engels, XJaM, Christian List, Toby~enwiki, Toby Bartels, Ryguasu, Hephaestos, Patrick,Michael Hardy,MartinHarper, TakuyaMurata, Eric119, Den fjättrade ankan~enwiki, Andres, Evercat, Renamed user 4, CharlesMatthews,Berteun, Dcoetzee, David Latapie, Dysprosia, Jitse Niesen, Krithin, Hyacinth, Spikey, Jeanmichel~enwiki, Flockmeal, Phil Boswell,Robbot, Sanders muc, Peak, Romanm, Gandalf61, Henrygb, Wikibot, Pengo, Tobias Bergemann, Adam78, Tosha, Giftlite, Dbenbenn,Vfp15, BenFrantzDale, Herbee, Fropuff, MichaelHaeckel, Macrakis, Python eggs, Rdsmith4, Mike Rosoft, Brianjd, Mormegil, Guan-abot, Paul August, Spearhead, EmilJ, BrokenSegue, Nortexoid, 3mta3, Obradovic Goran, Jonathunder, ABCD, Sligocki, Dzhim, Itsmine,HenryLi, Hq3473, Angr, Isnow, Qwertyus, MarSch, Salix alba, Bubba73, ChongDae, Salvatore Ingala, Chobot, YurikBot, RussBot,Rsrikanth05, Trovatore, Ms2ger, Saric, EtherealPurple, GrinBot~enwiki, TomMorris, SmackBot, InverseHypercube, Melchoir, FlashSh-eridan, Ohnoitsjamie, Joefaust, SMP, J. Spencer, Octahedron80, Iit bpd1962, Tamfang, Cybercobra, Dreadstar, RandomP, Jon Awbrey,Jóna Þórunn, Lambiam, Jim.belk, Vanished user v8n3489h3tkjnsdkq30u3f, Loadmaster, Hvn0413, Mets501, EdC~enwiki, Joseph Solisin Australia, Spindled, James pic, Amalas, Philiprbrenan, CBM, Gregbard, Cydebot, Pais, Julian Mendez, Malleus Fatuorum, Epbr123,Nick Number, Escarbot, Sluzzelin, .anacondabot, David Eppstein, Ttwo, Maurice Carbonaro, Ian.thomson, It Is Me Here, Daniel5Ko,NewEnglandYankee, DavidCBryant, VolkovBot, Zanardm, Rei-bot, Anonymous Dissident, Andy Dingley, SieBot, Niv.sarig, ToePeu.bot,Randomblue, Niceguyedc, Wounder, Nosolution182, Versus22, Palnot, AmeliaElizabeth, Feinoha, American Eagle, ThisIsMyWikipedi-aName, LaaknorBot, AnnaFrance, Numbo3-bot, Zorrobot, Legobot, Luckas-bot, Yobot, Ciphers, Xqbot, Nasnema, , GrouchoBot,LucienBOT, Pinethicket, Kiefer.Wolfowitz, Abductive, Jauhienij, FoxBot, Lotje, LilyKitty, Woodsy dong peep, EmausBot, Sharlack-Hames, Ystory, ClueBot NG, Cntras, Rezabot, Helpful Pixie Bot, Michael.croghan, Langing, Ugncreative Usergname, JYBot, Kephir,Tango303, Phinumu, Noyster, GeoffreyT2000, Skw27 and Anonymous: 82

• General topology Source: https://en.wikipedia.org/wiki/General_topology?oldid=674304842 Contributors: AxelBoldt, Eloquence, TobyBartels, AugPi, Revolver, Charles Matthews, Dysprosia, Jitse Niesen, Robbot, MathMartin, Tosha, Giftlite, Maximaximax, Rgdboer,Remuel, Tsirel, Msh210, Dallashan~enwiki, Sligocki, Brookie, Linas, BD2412, R.e.b., Mathbot, Dnwoodbury, Masnevets, YurikBot,Trovatore, Number 57, Googl, Reyk, Sardanaphalus, Bluebot, Nbarth, CBM, Chadnash, MER-C, The Transhumanist, DGG, VolkovBot,JohnBlackburne, YohanN7, JackSchmidt, Hans Adler, M.K.R.S. VEERA KUMAR, Addbot, Yobot, Sławomir Biały, Trappist the monk,TjBot, EmausBot, Fly by Night, ZéroBot, Anita5192, Oleg Viro, Brad7777, Majesty of Knowledge, ChrisGualtieri, APerson, Mogism,Aymankamelwiki, Brirush, Mark viking, Steynberg, Purnendu Karmakar, Nigellwh, K401sTL3 and Anonymous: 22

• Hausdorff space Source: https://en.wikipedia.org/wiki/Hausdorff_space?oldid=647054341 Contributors: AxelBoldt, Magnus Manske,Zundark, Tarquin, Toby Bartels, B4hand, Michael Hardy, JakeVortex, TakuyaMurata, Ellywa, Cyp, BenKovitz, Ideyal, Charles Matthews,Dcoetzee, Dysprosia, Grendelkhan, Fibonacci, Robbot, MathMartin, Bkell, Tobias Bergemann, Tosha, Giftlite, Fropuff, Icairns, Vasile,Clarknova, TedPavlic, Guanabot, Westendgirl, Paul August, Bender235, El C, Vipul, .:Ajvol:., Tsirel, Dallashan~enwiki, Eric Kvaalen,Sligocki, Isaac, Jim Slim, Oleg Alexandrov, Linas, StradivariusTV, Graham87, BD2412, Chobot, YurikBot, Wavelength, Hairy Dude,Jessesaurus, Bota47, DVDRW, Sardanaphalus, Nbarth, JonAwbrey, Germandemat, Mets501, Dp462090, Cydebot, Alazaris, Dharma6662000,W3asal, Arcresu, RobHar, Salgueiro~enwiki, JAnDbot, AntiSpamBot, Policron, TXiKiBoT, Broadbot, Ocsenave, SieBot, OKBot, Ran-domblue, EconomicsGuy, Kruusamägi, DumZiBoT, Addbot, Mortense, Lightbot, Luckas-bot, Yobot, Ciphers, Citation bot, Kenneth-leebaker, Sławomir Biały, Vectornaut, Lapasotka, EmausBot, Drusus 0, Helpful Pixie Bot, Brad7777, MikeHaskel, Jochen Burghardt,Hierarchivist and Anonymous: 44

• Neighbourhood (mathematics) Source: https://en.wikipedia.org/wiki/Neighbourhood_(mathematics)?oldid=682099720 Contributors:Patrick, Andres, Robbot, MathMartin, Giftlite, Fropuff, LiDaobing, Frenchwhale, Sam nead, PhotoBox, Paul August, Dmr2, El C,Obradovic Goran, Marc van Woerkom, Oleg Alexandrov, MFH, Salix alba, SpNeo, VKokielov, Mathbot, Kri, Hadaso, Hairy Dude,4C~enwiki, Akriasas, Cronholm144,Mets501, Dycedarg, Xantharius, Thijs!bot, Tchakra, VectorPosse, Salgueiro~enwiki, OM,Crazytonyi,Sullivan.t.j, David Eppstein, Kostisl, Maurice Carbonaro, Trumpet marietta 45750, Synthebot, Arcfrk, Bluestarlight37, SieBot, Bot-Multichill, Gerakibot, Yerpo, Svick, The Stickler, Addbot, Dkived, NjardarBot, LaaknorBot, Ciphers, ArthurBot, GrouchoBot, Yu V,D'ohBot, EmausBot, ZéroBot, SporkBot, ClueBot NG, Wcherowi, Helpful Pixie Bot, TricksterWolf, Stephan Kulla, Mark viking, KSFTand Anonymous: 27

• Open set Source: https://en.wikipedia.org/wiki/Open_set?oldid=664841828Contributors: AxelBoldt, Zundark, Taral, XJaM, Toby~enwiki,Toby Bartels, Miguel~enwiki, Patrick, Iulianu, Iorsh, Rob Hooft, Dino, Wikiborg, Dysprosia, Jitse Niesen, Robbot, MathMartin, Bkell,Saforrest, Tobias Bergemann, Giftlite, Lethe, Dratman, Jason Quinn, Python eggs, LiDaobing, Tzanko Matev, Moxfyre, Paul August,Elwikipedista~enwiki, Nickj, O18, Obradovic Goran, Nsaa, Gerweck, Dethron, Oleg Alexandrov, Simetrical, Mpatel, Magister Math-ematicae, Salix alba, Mike Segal, FlaBot, VKokielov, Margosbot~enwiki, Jrtayloriv, Chobot, Roboto de Ajvol, YurikBot, Hairy Dude,Bota47, Icedwater, Pred, Paul D. Anderson, Allens, Sardanaphalus, SmackBot, Incnis Mrsi, K-UNIT, Foxjwill, HLwiKi, Danpovey, LkN-sngth, Wizardman, Vina-iwbot~enwiki, Vanished user 9i39j3, AutomaticWriting, Jackzhp, Dslc, Xantharius, Dharma6662000, Thijs!bot,Tchakra, JAnDbot, Richard Giuly, Hashem62, Joeabauer, TomyDuby, Trumpet marietta 45750, Daniele.tampieri, Daniel5Ko, Policron,Dubhe.sk, Chrystomath, KylieTastic, VolkovBot, Camrn86, Cbigorgne, Jordankayla123, MartinPackerIBM, Ocsenave, SieBot, MiNom-breDeGuerra, Thobitz, Blacklemon67, Estirabot, Qwfp, Addbot, Wikomidia, Zorrobot, Legobot, EdwardLane, AnomieBOT, Ciphers,Jim1138, DannyAsher, Bdmy, Point-set topologist, RibotBOT, LivingBot, MarcelB612, Sheerun, Wham Bam Rock II, Grondilu, Erithil,ChuispastonBot, Mesoderm, MRG90, ChalkboardCowboy, Brirush, Horsegal1, Ganatuiyop and Anonymous: 58

• Separable space Source: https://en.wikipedia.org/wiki/Separable_space?oldid=648185468Contributors: AxelBoldt, Zundark, Toby Bar-tels, Michael Hardy, Ellywa, Charles Matthews, Robbot, MathMartin, Tobias Bergemann, Tosha, Giftlite, Lethe, Fropuff, Mbover-

62 CHAPTER 10. TOPOLOGY

load, Mbork~enwiki, Noisy, Rich Farmbrough, El C, Vipul, EmilJ, Denis.arnaud, Oleg Alexandrov, Guardian of Light, OdedSchramm,Marudubshinki, FlaBot, YurikBot, Hairy Dude, Hennobrandsma, SmackBot, Melchoir, DHN-bot~enwiki, Mets501, Dwmalone, Jamespic, Cydebot, Headbomb, Salgueiro~enwiki, JAnDbot, Sullivan.t.j, Polymedes~enwiki, Trumpetmarietta 45750, Daniele.tampieri, Plclark,Smaigol, JackSchmidt, Alexbot, Addbot, Topology Expert, SpBot, PV=nRT, Luckas-bot, Yobot, The Earwig, Hairer, Xqbot, Quarx314,FrescoBot, D'ohBot, DixonDBot, Setitup, WikitanvirBot, RDWD, Wgunther, ZéroBot, Zfeinst, ClueBot NG, DanGRV and Anonymous:25

• Subset Source: https://en.wikipedia.org/wiki/Subset?oldid=680865808 Contributors: Damian Yerrick, AxelBoldt, Youssefsan, XJaM,Toby Bartels, StefanRybo~enwiki, Edward, Patrick, TeunSpaans, Michael Hardy, Wshun, Booyabazooka, Ellywa, Oddegg, Andres,Charles Matthews, Timwi, Hyacinth, Finlay McWalter, Robbot, Romanm, Bkell, 75th Trombone, Tobias Bergemann, Tosha, Giftlite,Fropuff, Waltpohl, Macrakis, Tyler McHenry, SatyrEyes, Rgrg, Vivacissamamente, Mormegil, EugeneZelenko, Noisy, Deh, Paul Au-gust, Engmark, Spoon!, SpeedyGonsales, Obradovic Goran, Nsaa, Jumbuck, Raboof, ABCD, Sligocki, Mac Davis, Aquae, LFaraone,Chamaeleon, Firsfron, Isnow, Salix alba, VKokielov, Mathbot, Harmil, BMF81, Chobot, Roboto de Ajvol, YurikBot, Alpt, Dmharvey,KSmrq, NawlinWiki, Trovatore, Nick, Szhaider, Wasseralm, Sardanaphalus, Jacek Kendysz, BiT, Gilliam, Buck Mulligan, SMP, Or-angeDog, Bob K, Dreadstar, Bjankuloski06en~enwiki, Loadmaster, Vedexent, Amitch, Madmath789, Newone, CBM, Jokes Free4Me,345Kai, SuperMidget, Gregbard, WillowW, MC10, Thijs!bot, Headbomb, Marek69, RobHar, WikiSlasher, Salgueiro~enwiki, JAnDbot,.anacondabot, Pixel ;-), Pawl Kennedy, Emw, ANONYMOUS COWARD0xC0DE, RaitisMath, JCraw, Tgeairn, Ttwo, Maurice Car-bonaro, Acalamari, Gombang, NewEnglandYankee, Liatd41, VolkovBot, CSumit, Deleet, Rei-bot, AnonymousDissident, James.Spudeman,PaulTanenbaum, InformationSpace, Falcon8765, AlleborgoBot, P3d4nt, NHRHS2010, Garde, Paolo.dL, OKBot, Brennie8, Jons63,Loren.wilton, ClueBot, GorillaWarfare, PipepBot, The Thing That Should Not Be, DragonBot, Watchduck, Hans Adler, Computer97,Noosentaal, Versus22, PCHS-NJROTC, Andrew.Flock, Reverb123, Addbot, , Fyrael, PranksterTurtle, Numbo3-bot, Zorrobot, Jar-ble, JakobVoss, Luckas-bot, Yobot, Synchronism, AnomieBOT, Jim1138, Materialscientist, Citation bot, Martnym, NFD9001, Char-vest, 78.26, XQYZ, Egmontbot, Rapsar, HRoestBot, Suffusion of Yellow, Agent Smith (The Matrix), RenamedUser01302013, ZéroBot,Alexey.kudinkin, Chharvey, Quondum, Chewings72, 28bot, ClueBot NG, Wcherowi, Matthiaspaul, Bethre, Mesoderm, O.Koslowski,AwamerT, Minsbot, Pratyya Ghosh, YFdyh-bot, Ldfleur, ChalkboardCowboy, Saehry, Stephan Kulla, , Ilya23Ezhov, Sandshark23,Quenhitran, Neemasri, Prince Gull, Maranuel123, Alterseemann, Rahulmr.17, Michael James M, Johnkennethcfamero and Anonymous:185

• Topological space Source: https://en.wikipedia.org/wiki/Topological_space?oldid=683229082Contributors: AxelBoldt, Zundark, XJaM,Toby Bartels, Olivier, Patrick, Michael Hardy, Wshun, Kku, Dineshjk, Karada, Hashar, Zhaoway~enwiki, Revolver, Charles Matthews,Dcoetzee, Dysprosia, Kbk, Taxman, Phys, Robbot, Nizmogtr, Fredrik, Saaska, MathMartin, P0lyglut, Tobias Bergemann, Giftlite, GeneWard Smith, Lethe, Fropuff, Dratman, DefLog~enwiki, Rhobite, Luqui, Paul August, Dolda2000, Elwikipedista~enwiki, Tompw, Aude,SgtThroat, Tsirel, Marc van Woerkom, Varuna, Kuratowski’s Ghost, Msh210, Keenan Pepper, Danog, Sligocki, Spambit, Oleg Alexan-drov, Woohookitty, Graham87, BD2412, Grammarbot, FlaBot, Sunayana, Tillmo, Chobot, Algebraist, YurikBot, Wavelength, HairyDude, NawlinWiki, Rick Norwood, Bota47, Stefan Udrea, Hirak 99, Arthur Rubin, Lendu, JoanneB, Eaefremov, RonnieBrown, Sar-danaphalus, SmackBot, Maksim-e~enwiki, Sciyoshi~enwiki, DHN-bot~enwiki, Tsca.bot, Tschwenn, LkNsngth, Vriullop, Arialblack, Iri-descent, Devourer09,Mattbr, AndrewDelong, Kupirijo, Roccorossi, Xantharius, Thijs!bot, Konradek, Odoncaoa, Escarbot, Salgueiro~enwiki,JAnDbot, YK Times, Bkpsusmitaa, Jakob.scholbach, Bbi5291, Wdevauld, J.delanoy, Pharaoh of the Wizards, Maurice Carbonaro, TheMudge, Jmajeremy, Policron, TXiKiBoT, Anonymous Dissident, Plclark, Aaron Rotenberg, Jesin, Arcfrk, SieBot, MiNombreDeGuerra,JerroldPease-Atlanta, JackSchmidt, Failure.exe, Egmontaz, Palnot, SilvonenBot, Addbot, CarsracBot, AnnaFrance, ChenzwBot, Luckas-bot, Yobot, SwisterTwister, AnomieBOT, Ciphers, Materialscientist, Citation bot, DannyAsher, FlordiaSunshine342, J04n, Point-settopologist, FrescoBot, Jschnur, Jeroen De Dauw, TobeBot, Seahorseruler, Skakkle, Cstanford.math, ZéroBot, Chharvey, Wikfr, OrangeSuede Sofa, Liuthar, ClueBot NG, Wcherowi, Mesoderm, Vinícius Machado Vogt, Helpful Pixie Bot, Gaurav Nirala, Tom.hirschowitz,Pacerier, Solomon7968, Cpatra1984, Brad7777, Minsbot, LoganFromSA, MikeHaskel, Acer4666, Freeze S, Mark viking, Epicgenius,Kurt Artindagi, Improbable keeler, Amonk1962, GeoffreyT2000, KasparBot and Anonymous: 110

• Topology Source: https://en.wikipedia.org/wiki/Topology?oldid=683923685 Contributors: AxelBoldt, Brion VIBBER, Zundark, TheAnome, XJaM, Vanderesch, Toby~enwiki, Toby Bartels, Hhanke, Miguel~enwiki, Camembert, Hephaestos, Olivier, Bdesham, Patrick,Michael Hardy, Wshun, Liftarn, Gnomon42, Sannse, TakuyaMurata, GTBacchus, Karada, MightCould, CesarB, Cyp, Mark Foskey,Mxn, Darkov, Charles Matthews, Dcoetzee, Dino, Sicro, Dysprosia, Jitse Niesen, Wik, Zoicon5, Steinsky, Hyacinth, Saltine, Jean-michel~enwiki, Jusjih, Robbot, Mountain, Donreed, Altenmann, GeorgeKontopoulos, Gandalf61,MathMartin, Henrygb, Rursus, Robinh,Fuelbottle, TexasDex, Tobias Bergemann, Ramir, Pdenapo, Weialawaga~enwiki, Marc Venot, Tosha, Kevin Saff, Giftlite, Dbenbenn,Jyril, Rudolf 1922, Inter, Lethe, Fropuff, Everyking, Curps, Guanaco, Jason Quinn, Ptk~enwiki, Prosfilaes, Dan Gardner, Gadfium, Lu-casVB, Onco p53, APH,Maximaximax, Gauss, Abdull, ELApro, D6, Ta bu shi da yu, Imroy, Discospinster, Rich Farmbrough, LindsayH,Paul August, Dmr2, Violetriga, Gauge, Tompw, El C, Rgdboer, Aude, EmilJ, Keron Cyst, C S, Shenme, Jjk, Jung dalglish, Maurreen,Haham hanuka, Mdd, Varuna, Jumbuck, Msh210, Danog, Sligocki, Derbeth, Oleg Alexandrov, Brookie, Saeed, Velho, Woohookitty,Linas, Spamguy, Prophile, Oliphaunt, WadeSimMiser, Orz, MONGO, Dzordzm, Graham87, Magister Mathematicae, Porcher, Rjwilmsi,Mayumashu, Tangotango, Hychu, Salix alba, NonNobis~enwiki, Yamamoto Ichiro, FlaBot, Nivix, Isotope23, Windharp, Chobot, DylanThurston, Gdrbot, Algebraist, Wavelength, Borgx, Hairy Dude, Hede2000, Stephenb, Chaos, Cryptic, Rick Norwood, Joth, The Ogre,Trovatore, TechnoGuyRob, Crasshopper, Natkeeran, Aaron Schulz, EEMIV, Aidanb, PyroGamer, User27091, Stefan Udrea, Tetracube,Arthur Rubin, Bentong Isles, Naught101, Curpsbot-unicodify, Ilmari Karonen, TMott, RonnieBrown, Brentt, Sardanaphalus, SmackBot,RDBury, Mmernex, David Kernow, Honza Záruba, KnowledgeOfSelf, Delldot, Cokebingo, Alsandro, Wikikris, Gilliam, Betacommand,Skizzik, Chaojoker, Bluebot, The baron, MalafayaBot, Stevage, Darth Panda, Nick Levine, Alriode, Mhym, Lesnail, Jackohare, Ran-domP, Geoffr, Jon Awbrey, Sammy1339, Dr. Gabriel Gojon, Ohconfucius, Lambiam, Rory096, NongBot~enwiki, Atoll, Mr Stephen,Thevelho, Stephen B Streater, Jason.grossman, Madmath789, Francl, Jbolden1517, CRGreathouse, CmdrObot, Jrolland, CBM, Ranicki,Usgnus, Werratal, Myasuda, Yaris678, Cydebot, Gogo Dodo, Corpx, Dr.enh, Starship Trooper, Gaoos, Mariontte User, Dharma6662000,Thijs!bot, Epbr123, J. Charles Taylor, Knakts, Kilva, Perrygogas, JustAGal, Chadnash, AbcXyz, Escarbot, LachlanA, AntiVandalBot,Seaphoto, Tchakra, Emeraldcityserendipity, Weixifan, Lfstevens, Byrgenwulf, Ioeth, GromXXVII, Turgidson, MER-C, Skomorokh,Magioladitis, Jéské Couriano, SwiftBot, Bubba hotep, Ensign beedrill, Sullivan.t.j, SnakeChess5, Pax:Vobiscum, Rootneg2, MartinBot,Erkan Yilmaz, J.delanoy, Maurice Carbonaro, MarcoLittel, Policron, Jamesofur, Alan U. Kennington, Jeff G., JohnBlackburne, Nouser-namesleft, Topologyxpert, TXiKiBoT, Ttopperr, Philosotox, Anonymous Dissident, Qxz, Digby Tantrum, Softkitten88, ARUNKUMARP.R,Wolfrock, Enviroboy, Dmcq, MiamiMath, Symane, ADOGisAboat, Katzmik, Rknasc, Rybu, YohanN7, SieBot, Ivan Štambuk, Mel-dor, Triwbe, Aristolaos, Nicinic, Pendlehaven, Daniarmo, MiNombreDeGuerra, JorgenW, Kumioko, Valeria.depaiva, Vituzzu, Stfg, Lau-rentseries, Jludwig, ClueBot, The Thing That Should Not Be, Stevanspringer, TheSmuel, Monty42, SapphireJay, SchreiberBike, Triath-ematician, Manatee331, Robertabrams, Novjunulo, Fastily, Pi.C.Noizecehx, ErickOrtiz, Tilmanbauer, MystBot, Addbot, Some jerk on

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64 CHAPTER 10. TOPOLOGY

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