Separation Axioms

Separation axiomsFrom Wikipedia, the free encyclopedia

Contents

1 Collectionwise normal space 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Dowker space 22.1 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Hausdor space 33.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.5 Preregularity versus regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.6 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.7 Algebra of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.8 Academic humour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 History of the separation axioms 74.1 Origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 Dierent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.3 Completely Hausdor, Urysohn, and T spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Kolmogorov space 95.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.2 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.2.1 Spaces which are not T0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.2.2 Spaces which are T0 but not T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5.3 Operating with T0 spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.4 The Kolmogorov quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.5 Removing T0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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5.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 Monotonically normal space 126.1 Equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

6.1.1 Denition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.1.2 Denition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.1.3 Denition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.3 Some discussion links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7 Normal space 147.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.2 Examples of normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.3 Examples of non-normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.5 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.6 Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8 Paracompact space 178.1 Paracompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.4 Paracompact Hausdor Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

8.4.1 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.5 Relationship with compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8.5.1 Comparison of properties with compactness . . . . . . . . . . . . . . . . . . . . . . . . . 208.6 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8.6.1 Denition of relevant terms for the variations . . . . . . . . . . . . . . . . . . . . . . . . . 218.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

9 Regular space 239.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.2 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.3 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.4 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

10 Semiregular space 2610.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

CONTENTS iii

11 Separation axiom 2711.1 Preliminary denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.2 Main denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.3 Relationships between the axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.4 Other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

12 Sober space 3312.1 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

13 T1 space 3513.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.4 Generalisations to other kinds of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3713.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

14 Topological indistinguishability 3814.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3814.2 Specialization preorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3914.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

14.3.1 Equivalent conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3914.3.2 Equivalence classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.3.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

14.4 Kolmogorov quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

15 Tychono space 4215.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4215.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4215.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

15.3.1 Preservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4315.3.2 Real-valued continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4315.3.3 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.3.4 Compactications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.3.5 Uniform structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

15.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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16 Urysohn and completely Hausdor spaces 4516.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.2 Naming conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.3 Relation to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

17 Urysohns lemma 4717.1 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4717.2 Sketch of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4717.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4817.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4817.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 49

17.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4917.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5017.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Chapter 1

Collectionwise normal space

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

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

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Chapter 2

Dowker space

In the mathematical eld of general topology, a Dowker space is a topological space that is T4 but not countablyparacompact. They are named after Cliord Hugh Dowker.The non-trivial task of providing an example of a Dowker space (and therefore also proving their existence as math-ematical objects) helped mathematicians better understand the nature and variety of topological spaces. Topologicalspaces are sets together with some subsets (designated as open sets) satisfying certain properties. Topologicalspaces arose as generalization of the open sets of spaces studied in elementary mathematics, such as open disks inthe Euclidean plane, open balls in the Euclidean space, and open intervals of the real line.

2.1 EquivalencesDowker showed, in 1951, the following:If X is a normal T1 space (that is, a T4 space), then the following are equivalent:

X is a Dowker space The product of X with the unit interval is not normal.[1]

X is not countably metacompact.

Dowker conjectured that there were no Dowker spaces, and the conjecture was not resolved until M. E. Rudin con-structed one[2] in 1971. Rudins counterexample is a very large space (of cardinality @@0! ). Zoltn Balogh gave therst ZFC construction[3] of a small (cardinality continuum) example, which was more well-behaved than Rudins.Using PCF theory, M. Kojman and S. Shelah constructed[4] a subspace of Rudins Dowker space of cardinality @!+1that is also Dowker.

2.2 References[1] Dowker, C. H. (1951). On countably paracompact spaces (PDF). Can. J. Math. 3: 219224. doi:10.4153/CJM-1951-

026-2. Zbl 0042.41007. Retrieved March 29, 2015.

[2] Rudin, Mary Ellen (1971). A normal space X for which X I is not normal (PDF). Fundam. Math. (Polish Academyof Sciences) 73 (2): 179186. Zbl 0224.54019. Retrieved March 29, 2015.

[3] Balogh, Zoltan T. (August 1996). A small Dowker space in ZFC (PDF). Proc. Amer. Math. Soc. 124 (8): 25552560.Zbl 0876.54016. Retrieved March 29, 2015.

[4] Kojman, Menachem; Shelah, Saharon (1998). A ZFC Dowker space in @!+1 : an application of PCF theory to topology(PDF). Proc. Amer. Math. Soc. (American Mathematical Society) 126 (8): 24592465. Retrieved March 29, 2015.

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Chapter 3

Hausdor space

In topology and related branches of mathematics, a Hausdor 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 Hausdor condition (T2) is the most frequently used and discussed. It implies the uniquenessof limits of sequences, nets, and lters.Hausdor spaces are named after Felix Hausdor, one of the founders of topology. Hausdors original denitionof a topological space (in 1914) included the Hausdor condition as an axiom.

3.1 Denitions

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 Hausdor 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 Hausdor 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 Hausdor 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

3

4 CHAPTER 3. HAUSDORFF SPACE

Hausdor.

3.2 EquivalencesFor a topological space X, the following are equivalent:

X is a Hausdor space. Limits of nets in X are unique.[1]

Limits of lters 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 = {(x,x) | x X} is closed as a subset of the product space X X.

3.3 Examples and counterexamplesAlmost all spaces encountered in analysis are Hausdor; most importantly, the real numbers (under the standardmetric topology on real numbers) are a Hausdor space. More generally, all metric spaces are Hausdor. In fact,many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdor conditionexplicitly stated in their denitions.A simple example of a topology that is T1 but is not Hausdor is the conite topology dened on an innite set.Pseudometric spaces typically are not Hausdor, but they are preregular, and their use in analysis is usually only in theconstruction of Hausdor gauge spaces. Indeed, when analysts run across a non-Hausdor space, it is still probablyat least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdor.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 Hausdor.While the existence of unique limits for convergent nets and lters implies that a space is Hausdor, there are non-Hausdor T1 spaces in which every convergent sequence has a unique limit.[4]

3.4 PropertiesSubspaces and products of Hausdor spaces are Hausdor,[5] but quotient spaces of Hausdor spaces need not beHausdor. In fact, every topological space can be realized as the quotient of some Hausdor space.[6]

Hausdor spaces are T1, meaning that all singletons are closed. Similarly, preregular spaces are R0.Another nice property of Hausdor spaces is that compact sets are always closed.[7] This may fail in non-Hausdorspaces such as Sierpiski space.The denition of a Hausdor space says that points can be separated by neighborhoods. It turns out that this impliessomething which is seemingly stronger: in a Hausdor 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 satisfyUrysohns lemma and the Tietze extension theorem and have partitions of unity subordinate to locally nite opencovers. The Hausdor versions of these statements are: every locally compact Hausdor space is Tychono, andevery compact Hausdor space is normal Hausdor.The following results are some technical properties regarding maps (continuous and otherwise) to and fromHausdorspaces.

3.5. PREREGULARITY VERSUS REGULARITY 5

Let f : X Y be a continuous function and suppose Y is Hausdor. Then the graph of f, f(x; f(x)) j x 2 Xg , isa closed subset of X Y.Let f : X Y be a function and let ker(f) , f(x; x0) j f(x) = f(x0)g be its kernel regarded as a subspace of X X.

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

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

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

If f,g : X Y are continuous maps and Y is Hausdor then the equalizer eq(f; g) = fx j f(x) = g(x)g is closed inX. It follows that if Y is Hausdor and f and g agree on a dense subset of X then f = g. In other words, continuousfunctions into Hausdor spaces are determined by their values on dense subsets.Let f : X Y be a closed surjection such that f1(y) is compact for all y Y. Then if X is Hausdor so is Y.Let f : X Y be a quotient map with X a compact Hausdor space. Then the following are equivalent

Y is Hausdor

f is a closed map

ker(f) is closed

3.5 Preregularity versus regularityAll regular spaces are preregular, as are all Hausdor spaces. There are many results for topological spaces that holdfor both regular and Hausdor spaces. Most of the time, these results hold for all preregular spaces; they were listedfor regular and Hausdor 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 Hausdor spaces.There are many situations where another condition of topological spaces (such as paracompactness or local com-pactness) will imply regularity if preregularity is satised. Such conditions often come in two versions: a regularversion and a Hausdor version. Although Hausdor spaces aren't generally regular, a Hausdor space that is also(say) locally compact will be regular, because any Hausdor space is preregular. Thus from a certain point of view,it is really preregularity, rather than regularity, that matters in these situations. However, denitions 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.

3.6 VariantsThe terms Hausdor, 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 lters (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 Hausdor condi-tion in these cases reduces to the T0 condition. These are also the spaces in which completeness makes sense, andHausdorness is a natural companion to completeness in these cases. Specically, a space is complete if and only ifevery Cauchy net has at least one limit, while a space is Hausdor if and only if every Cauchy net has at most onelimit (since only Cauchy nets can have limits in the rst place).

6 CHAPTER 3. HAUSDORFF SPACE

3.7 Algebra of functionsThe algebra of continuous (real or complex) functions on a compact Hausdor space is a commutative C*-algebra,and conversely by the BanachStone 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.

3.8 Academic humour Hausdor condition is illustrated by the pun that in Hausdor spaces any two points can be housed o fromeach other by open sets.[9]

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

3.9 See also Quasitopological space Weak Hausdor space Fixed-point space, a Hausdor space X such that every continuous function f:XX has a xed point.

3.10 Notes[1] Willard, pp. 8687.

[2] Willard, pp. 8687.

[3] Bourbaki, p. 75.

[4] van Douwen, Eric K. (1993). An anti-Hausdor Frchet space in which convergent sequences have unique limits.Topology and its Applications 51 (2): 147158. doi:10.1016/0166-8641(93)90147-6.

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

[6] Shimrat, M. (1956). Decomposition spaces and separation properties. Quart. J. Math. 2: 128129.

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

[8] Willard, p. 124.

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

3.11 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), Hausdor space, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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

Chapter 4

History of the separation axioms

The history of the separation axioms in general topology has been convoluted, with many meanings competing forthe same terms and many terms competing for the same concept.

4.1 OriginsBefore the current general denition of topological space, there weremany denitions oered, some of which assumed(what we now think of as) some separation axioms. For example, the denition given by Felix Hausdor in 1914 isequivalent to the modern denition plus the Hausdor separation axiom.The separation axioms, as a group, became important in the study of metrisability: the question of which topologicalspaces can be given the structure of a metric space. Metric spaces satisfy all of the separation axioms; but in fact,studying spaces that satisfy only some axioms helps build up to the notion of full metrisability.The separation axioms that were rst studied together in this way were the axioms for accessible spaces, Hausdorspaces, regular spaces, and normal spaces. Topologists assigned these classes of spaces the names T1, T2, T3, andT4. Later this system of numbering was extended to include T0, T, T (or T), T5, and T6.But this sequence had its problems. The idea was supposed to be that every Ti space is a special kind of Tj space if i> j. But this is not necessarily true, as denitions vary. For example, a regular space (called T3) does not have to bea Hausdor space (called T2), at least not according to the simplest denition of regular spaces.

4.2 Dierent denitionsEvery author agreed on T0, T1, and T2. For the other axioms, however, dierent authors could use signicantlydierent denitions, depending on what they were working on. These dierences could develop because, if oneassumes that a topological space satises the T1 axiom, then the various denitions are (in most cases) equivalent.Thus, if one is going to make that assumption, then one would want to use the simplest denition. But if one did notmake that assumption, then the simplest denition might not be the right one for the most useful concept; in any case,it would destroy the (transitive) entailment of Ti by Tj, allowing (for example) non-Hausdor regular spaces.Topologists working on the metrisation problem generally did assume T1; after all, all metric spaces are T1. Thus,they used the simplest denitions for the Ti. Then, for those occasions when they did not assume T1, they used words(regular and normal) for the more complicated denitions, in order to contrast them with the simpler ones. Thisapproach was used as late as 1970 with the publication of Counterexamples in Topology by Lynn A. Steen and J.Arthur Seebach, Jr.In contrast, general topologists, led by John L. Kelley in 1955, usually did not assume T1, so they studied the separationaxioms in the greatest generality from the beginning. They used the more complicated denitions for Ti, so that theywould always have a nice property relating Ti to Tj. Then, for the simpler denitions, they used words (again, regularand normal). Both conventions could be said to follow the original meanings; the dierent meanings are the samefor T1 spaces, which was the original context. But the result was that dierent authors used the various terms inprecisely opposite ways. Adding to the confusion, some literature will observe a nice distinction between an axiom

7

8 CHAPTER 4. HISTORY OF THE SEPARATION AXIOMS

and the space that satises the axiom, so that a T3 space might need to satisfy the axioms T3 and T0 (e.g., in theEncyclopedic Dictionary of Mathematics, 2nd ed.).Since 1970, the general topologists terms have been growing in popularity, including in other branches of mathe-matics, such as analysis. (Thus we use their terms in Wikipedia.) But usage is still not consistent.

4.3 Completely Hausdor, Urysohn, and T spacesMain article: Completely Hausdor space

Steen and Seebach dene a Urysohn space as a space with a Urysohn function for any two points. Willard callsthis a completely Hausdor space. Steen & Seebach dene a completely Hausdor space or T space as a space inwhich every two points are separated by closed neighborhoods, which Willard calls a Urysohn space or T space.(Wikipedia follows Willard.)

4.4 References John L. Kelley; General Topology; ISBN 0-387-90125-6 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

Stephen Willard, General Topology, Addison-Wesley, 1970. Reprinted by Dover Publications, New York,2004. ISBN 0-486-43479-6 (Dover edition).

Chapter 5

Kolmogorov space

In topology and related branches of mathematics, a topological space X is a T0 space orKolmogorov space (namedafter Andrey Kolmogorov) if for every pair of distinct points of X, at least one of them has an open neighborhoodnot containing the other. In a T0 space all points are topologically distinguishable.This condition, called the T0 condition, is the weakest of the separation axioms. Nearly all topological spacesnormally studied in mathematics are T0 spaces. In particular, all T1 spaces, i.e., all spaces in which for every pair ofdistinct points each has a neighborhood not containing the other, are T0 spaces. This includes all T2 (or Hausdor)spaces, i.e., all topological spaces in which distinct points have disjoint neighbourhoods. Given any topological spaceone can construct a T0 space by identifying topologically indistinguishable points.T0 spaces that are not T1 spaces are exactly those spaces for which the specialization preorder is a nontrivial partialorder. Such spaces naturally occur in computer science, specically in denotational semantics.

5.1 DenitionA T0 space is a topological space in which every pair of distinct points is topologically distinguishable. That is, forany two dierent points x and y there is an open set which contains one of these points and not the other.Note that topologically distinguishable points are automatically distinct. On the other hand, if the singleton sets {x}and {y} are separated, then the points x and y must be topologically distinguishable. That is,

separated topologically distinguishable distinct

The property of being topologically distinguishable is, in general, stronger than being distinct but weaker than beingseparated. In a T0 space, the second arrow above reverses; points are distinct if and only if they are distinguishable.This is how the T0 axiom ts in with the rest of the separation axioms.

5.2 Examples and nonexamplesNearly all topological spaces normally studied in mathematics are T0. In particular, all Hausdor (T2) spaces and T1spaces are T0.

5.2.1 Spaces which are not T0 A set with more than one element, with the trivial topology. No points are distinguishable.

The set R2 where the open sets are the Cartesian product of an open set in R and R itself, i.e., the producttopology of R with the usual topology and R with the trivial topology; points (a,b) and (a,c) are not distin-guishable.

9

10 CHAPTER 5. KOLMOGOROV SPACE

The space of all measurable functions f from the real line R to the complex plane C such that the Lebesgueintegral of |f(x)|2 over the entire real line is nite. Two functions which are equal almost everywhere areindistinguishable. See also below.

5.2.2 Spaces which are T0 but not T1 The Zariski topology on Spec(R), the prime spectrum of a commutative ring R is always T0 but generallynot T1. The non-closed points correspond to prime ideals which are not maximal. They are important to theunderstanding of schemes.

The particular point topology on any set with at least two elements is T0 but not T1 since the particular pointis not closed (its closure is the whole space). An important special case is the Sierpiski space which is theparticular point topology on the set {0,1}.

The excluded point topology on any set with at least two elements is T0 but not T1. The only closed point isthe excluded point.

The Alexandrov topology on a partially ordered set is T0 but will not be T1 unless the order is discrete (agreeswith equality). Every nite T0 space is of this type. This also includes the particular point and excluded pointtopologies as special cases.

The right order topology on a totally ordered set is a related example. The overlapping interval topology is similar to the particular point topology since every open set includes 0. Quite generally, a topological space X will be T0 if and only if the specialization preorder on X is a partialorder. However, X will be T1 if and only if the order is discrete (i.e. agrees with equality). So a space will beT0 but not T1 if and only if the specialization preorder on X is a non-discrete partial order.

5.3 Operating with T0 spacesExamples of topological space typically studied are T0. Indeed, whenmathematicians in many elds, notably analysis,naturally run across non-T0 spaces, they usually replace them with T0 spaces, in a manner to be described below.To motivate the ideas involved, consider a well-known example. The space L2(R) is meant to be the space of allmeasurable functions f from the real line R to the complex plane C such that the Lebesgue integral of |f(x)|2 over theentire real line is nite. This space should become a normed vector space by dening the norm ||f || to be the squareroot of that integral. The problem is that this is not really a norm, only a seminorm, because there are functionsother than the zero function whose (semi)norms are zero. The standard solution is to dene L2(R) to be a set ofequivalence classes of functions instead of a set of functions directly. This constructs a quotient space of the originalseminormed vector space, and this quotient is a normed vector space. It inherits several convenient properties fromthe seminormed space; see below.In general, when dealing with a xed topology T on a set X, it is helpful if that topology is T0. On the other hand,when X is xed but T is allowed to vary within certain boundaries, to force T to be T0 may be inconvenient, sincenon-T0 topologies are often important special cases. Thus, it can be important to understand both T0 and non-T0versions of the various conditions that can be placed on a topological space.

5.4 The Kolmogorov quotientTopological indistinguishability of points is an equivalence relation. No matter what topological space X might beto begin with, the quotient space under this equivalence relation is always T0. This quotient space is called theKolmogorov quotient of X, which we will denote KQ(X). Of course, if X was T0 to begin with, then KQ(X) andX are naturally homeomorphic. Categorically, Kolmogorov spaces are a reective subcategory of topological spaces,and the Kolmogorov quotient is the reector.Topological spacesX and Y areKolmogorov equivalentwhen their Kolmogorov quotients are homeomorphic. Manyproperties of topological spaces are preserved by this equivalence; that is, if X and Y are Kolmogorov equivalent, thenX has such a property if and only if Y does. On the other hand, most of the other properties of topological spaces imply

5.5. REMOVING T0 11

T0-ness; that is, if X has such a property, then Xmust be T0. Only a few properties, such as being an indiscrete space,are exceptions to this rule of thumb. Even better, many structures dened on topological spaces can be transferredbetween X and KQ(X). The result is that, if you have a non-T0 topological space with a certain structure or property,then you can usually form a T0 space with the same structures and properties by taking the Kolmogorov quotient.The example of L2(R) displays these features. From the point of view of topology, the seminormed vector space thatwe started with has a lot of extra structure; for example, it is a vector space, and it has a seminorm, and these denea pseudometric and a uniform structure that are compatible with the topology. Also, there are several properties ofthese structures; for example, the seminorm satises the parallelogram identity and the uniform structure is complete.The space is not T0 since any two functions in L2(R) which are equal almost everywhere are indistinguishable with thistopology. When we form the Kolmogorov quotient, the actual L2(R), these structures and properties are preserved.Thus, L2(R) is also a complete seminormed vector space satisfying the parallelogram identity. But we actually get abit more, since the space is now T0. A seminorm is a norm if and only if the underlying topology is T0, so L2(R) isactually a complete normed vector space satisfying the parallelogram identity otherwise known as a Hilbert space.And it is a Hilbert space that mathematicians (and physicists, in quantum mechanics) generally want to study. Notethat the notation L2(R) usually denotes the Kolmogorov quotient, the set of equivalence classes of square integrablefunctions which dier on sets of measure zero, rather than simply the vector space of square integrable functionswhich the notation suggests.

5.5 Removing T0Although norms were historically dened rst, people came up with the denition of seminorm as well, which is asort of non-T0 version of a norm. In general, it is possible to dene non-T0 versions of both properties and structuresof topological spaces. First, consider a property of topological spaces, such as being Hausdor. One can then deneanother property of topological spaces by dening the space X to satisfy the property if and only if the Kolmogorovquotient KQ(X) is Hausdor. This is a sensible, albeit less famous, property; in this case, such a space X is calledpreregular. (There even turns out to be a more direct denition of preregularity). Now consider a structure that canbe placed on topological spaces, such as a metric. We can dene a new structure on topological spaces by letting anexample of the structure on X be simply a metric on KQ(X). This is a sensible structure on X; it is a pseudometric.(Again, there is a more direct denition of pseudometric.)In this way, there is a natural way to remove T0-ness from the requirements for a property or structure. It is generallyeasier to study spaces that are T0, but it may also be easier to allow structures that aren't T0 to get a fuller picture.The T0 requirement can be added or removed arbitrarily using the concept of Kolmogorov quotient.

5.6 External links History of weak separation axioms (PDF le)

Chapter 6

Monotonically normal space

In mathematics, amonotonically normal space is a particular kind of normal space, with some special characteris-tics, and is such that it is hereditarily normal, and any two separated subsets are strongly separated. They are denedin terms of a monotone normality operator.A T1 topological space (X; T ) is said to be monotonically normal if the following condition holds:For every x 2 G , where G is open, there is an open set (x;G) such that

1. x 2 (x;G) G2. if (x;G) \ (y;H) 6= ; then either x 2 H or y 2 G .

There are some equivalent criteria of monotone normality.

6.1 Equivalent denitions

6.1.1 Denition 2A space X is called monotonically normal if it is T1 and for each pair of disjoint closed subsets A;B there is an openset G(A;B) with the properties

1. A G(A;B) G(A;B) XnB and2. G(A;B) G(A0; B0) , whenever A A0 and B0 B .

This operator G is called monotone normality operator.Note that if G is a monotone normality operator, then ~G dened by ~G(A;B) = G(A;B)nG(B;A) is also amonotone normality operator; and ~G satises

~G(A;B) \ ~G(B;A) = ;For this reason we some time take the monotone normality operator so as to satisfy the above requirement; and thatfacilitates the proof of some theorems and of the equivalence of the denitions as well.

6.1.2 Denition 3A space X is called monotonically normal if it is T1 ,and to each pair (A, B) of subsets of X, with A \ B = ; =B \A , one can assign an open subset G(A, B) of X such that

1. A G(A;B) G(A;B) XnB;2. G(A;B) G(A0; B0) whenever A A0 and B0 B .

12

6.2. PROPERTIES 13

6.1.3 Denition 4A space X is called monotonically normal if it is T1 and there is a function H that assigns to each ordered pair (p,C)where C is closed and p is without C, an open set H(p,C) satisfying:

1. p 2 H(p; C) XnC2. if D is closed and p 62 C D thenH(p; C) H(p;D)3. if p 6= q are points in X, then H(p; fqg) \H(q; fpg) = ; .

6.2 PropertiesAn important example of these spaces would be, assuming Axiom of Choice, the linearly ordered spaces; however,it really needs axiom of choice for an arbitrary linear order to be normal (see van Douwens paper). Any generalisedmetric is monotonically normal even without choice. An important property of monotonically normal spaces is thatany two separated subsets are strongly separated there. Monotone normality is hereditary property and a monotoni-cally normal space is always normal by the rst condition of the second equivalent denition.We list up some of the properties :

1. A closed map preserves monotone normality.2. A monotonically normal space is hereditarily collectionwise normal.

3. Elastic spaces are monotonically normal.

6.3 Some discussion links Heath, R. W.; Lutzer, D. J.; Zenor, P. L. (April 1973). Monotonically Normal Spaces (PDF). Transactionsof the American Mathematical Society 178: 481493.

Borges, Carlos R. (March 1973). A Study of Monotonically Normal Spaces (PDF). Proceedings of theAmerican Mathematical Society 38 (1): 211214.

van Douwen, Eric K. (September 1985). Horrors of Topology Without AC: A Nonnormal Orderable Space(PDF). Proceedings of the American Mathematical Society 95 (1): 101105.

Gartside, P. M. (1997). Cardinal Invariants of Monotonically Normal Spaces (PDF). Topology and its Ap-plications 77 (3): 303314.

Henno Brandsmas discussion about Monotone Normality in Topology Atlas can be viewed here

Chapter 7

Normal space

For normal vector space, see normal (geometry).

In topology and related branches of mathematics, a normal space is a topological space X that satises Axiom T4:every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdor space is also called aT4 space. These conditions are examples of separation axioms and their further strengthenings dene completelynormal Hausdor spaces, or T5 spaces, and perfectly normal Hausdor spaces, or T6 spaces.

7.1 DenitionsA topological space X is a normal space if, given any disjoint closed sets E and F, there are open neighbourhoodsU of E and V of F that are also disjoint. More intuitively, this condition says that E and F can be separated byneighbourhoods.

U

E

V

F

The closed sets E and F, here represented by closed disks on opposite sides of the picture, are separated by their respective neigh-bourhoods U and V, here represented by larger, but still disjoint, open disks.

A T4 space is a T1 space X that is normal; this is equivalent to X being normal and Hausdor.A completely normal space or a hereditarily normal space is a topological space X such that every subspace of Xwith subspace topology is a normal space. It turns out that X is completely normal if and only if every two separatedsets can be separated by neighbourhoods.A completely T4 space, or T5 space is a completely normal T1 space topological space X, which implies that X is

14

7.2. EXAMPLES OF NORMAL SPACES 15

Hausdor; equivalently, every subspace of X must be a T4 space.A perfectly normal space is a topological space X in which every two disjoint closed sets E and F can be preciselyseparated by a continuous function f fromX to the real lineR: the preimages of {0} and {1} under f are, respectively,E and F. (In this denition, the real line can be replaced with the unit interval [0,1].)It turns out that X is perfectly normal if and only if X is normal and every closed set is a G set. Equivalently, X isperfectly normal if and only if every closed set is a zero set. Every perfectly normal space is automatically completelynormal.[1]

A Hausdor perfectly normal space X is a T6 space, or perfectly T4 space.Note that the terms normal space and T4" and derived concepts occasionally have a dierent meaning. (Nonethe-less, T5" always means the same as completely T4", whatever that may be.) The denitions given here are the onesusually used today. For more on this issue, see History of the separation axioms.Terms like normal regular space" and normal Hausdor space also turn up in the literature they simply meanthat the space both is normal and satises the other condition mentioned. In particular, a normal Hausdor space isthe same thing as a T4 space. Given the historical confusion of the meaning of the terms, verbal descriptions whenapplicable are helpful, that is, normal Hausdor instead of T4", or completely normal Hausdor instead of T5".Fully normal spaces and fully T4 spaces are discussed elsewhere; they are related to paracompactness.A locally normal space is a topological space where every point has an open neighbourhood that is normal. Everynormal space is locally normal, but the converse is not true. A classical example of a completely regular locallynormal space that is not normal is the Nemytskii plane.

7.2 Examples of normal spacesMost spaces encountered in mathematical analysis are normal Hausdor spaces, or at least normal regular spaces:

All metric spaces (and hence all metrizable spaces) are perfectly normal Hausdor; All pseudometric spaces (and hence all pseudometrisable spaces) are perfectly normal regular, although not ingeneral Hausdor;

All compact Hausdor spaces are normal; In particular, the Stoneech compactication of a Tychono space is normal Hausdor; Generalizing the above examples, all paracompact Hausdor spaces are normal, and all paracompact regularspaces are normal;

All paracompact topological manifolds are perfectly normal Hausdor. However, there exist non-paracompactmanifolds which are not even normal.

All order topologies on totally ordered sets are hereditarily normal and Hausdor. Every regular second-countable space is completely normal, and every regular Lindelf space is normal.

Also, all fully normal spaces are normal (even if not regular). Sierpinski space is an example of a normal space thatis not regular.

7.3 Examples of non-normal spacesAn important example of a non-normal topology is given by the Zariski topology on an algebraic variety or on thespectrum of a ring, which is used in algebraic geometry.A non-normal space of some relevance to analysis is the topological vector space of all functions from the real lineR to itself, with the topology of pointwise convergence. More generally, a theorem of A. H. Stone states that theproduct of uncountably many non-compact metric spaces is never normal.

16 CHAPTER 7. NORMAL SPACE

7.4 PropertiesEvery closed subset of a normal space is normal. The continuous and closed image of a normal space is normal.[2]

The main signicance of normal spaces lies in the fact that they admit enough continuous real-valued functions, asexpressed by the following theorems valid for any normal space X.Urysohns lemma: If A and B are two disjoint closed subsets of X, then there exists a continuous function f from Xto the real line R such that f(x) = 0 for all x in A and f(x) = 1 for all x in B. In fact, we can take the values of f to beentirely within the unit interval [0,1]. (In fancier terms, disjoint closed sets are not only separated by neighbourhoods,but also separated by a function.)More generally, the Tietze extension theorem: If A is a closed subset of X and f is a continuous function from A toR, then there exists a continuous function F: X R which extends f in the sense that F(x) = f(x) for all x in A.If U is a locally nite open cover of a normal space X, then there is a partition of unity precisely subordinate to U.(This shows the relationship of normal spaces to paracompactness.)In fact, any space that satises any one of these conditions must be normal.A product of normal spaces is not necessarily normal. This fact was rst proved by Robert Sorgenfrey. An example ofthis phenomenon is the Sorgenfrey plane. Also, a subset of a normal space need not be normal (i.e. not every normalHausdor space is a completely normal Hausdor space), since every Tychono space is a subset of its Stoneechcompactication (which is normal Hausdor). A more explicit example is the Tychono plank.

7.5 Relationships to other separation axiomsIf a normal space is R0, then it is in fact completely regular. Thus, anything from normal R0" to normal completelyregular is the same as what we normally call normal regular. Taking Kolmogorov quotients, we see that all normalT1 spaces are Tychono. These are what we normally call normal Hausdor spaces.A topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, thereare disjoint open sets containing them. Every normal space is pseudonormal, but not vice versa.Counterexamples to some variations on these statements can be found in the lists above. Specically, Sierpinski spaceis normal but not regular, while the space of functions from R to itself is Tychono but not normal.

7.6 Citations[1] Munkres 2000, p. 213

[2] Willard, Stephen (1970). General topology. Reading, Mass.: Addison-Wesley Pub. Co. pp. 100101. ISBN 0486434796.

7.7 References Kemoto, Nobuyuki (2004). Higher Separation Axioms. In K.P. Hart, J. Nagata, and J.E. Vaughan. Ency-clopedia of General Topology. Amsterdam: Elsevier Science. ISBN 0-444-50355-2.

Munkres, James R. (2000). Topology (2nd ed.). Prentice-Hall. ISBN 0-13-181629-2. Sorgenfrey, R.H. (1947). On the topological product of paracompact spaces. Bull. Amer. Math. Soc. 53:631632. doi:10.1090/S0002-9904-1947-08858-3.

Stone, A. H. (1948). Paracompactness and product spaces. Bull. Amer. Math. Soc. 54: 977982.doi:10.1090/S0002-9904-1948-09118-2.

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

Chapter 8

Paracompact space

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

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

X [2A

U:

A cover of a topological space X is open if all its members are open sets. A renement of a cover of a space X is anew cover of the same space such that every set in the new cover is a subset of some set in the old cover. In symbols,the cover V = {V : in B} is a renement of the cover U = {U : in A} if and only if, for any V in V, thereexists some U in U such that VU.An open cover of a space X is locally nite if every point of the space has a neighborhood that intersects only nitelymany sets in the cover. In symbols, U = {U : in A} is locally nite if and only if, for any x in X, there exists someneighbourhood V(x) of x such that the set

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

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

8.2 Examples Every compact space is paracompact.

17

18 CHAPTER 8. PARACOMPACT SPACE

Every regular Lindelf space is paracompact. In particular, every locally compact Hausdor second-countablespace is paracompact.

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

Every CW complex is paracompact [1]

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

Some examples of spaces that are not paracompact include:

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

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

The Prfer manifold is a non-paracompact surface. The bagpipe theorem shows that there are 21 isomorphism classes of non-paracompact surfaces.

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

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

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

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

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

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

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

8.4 Paracompact Hausdor SpacesParacompact spaces are sometimes required to also be Hausdor to extend their properties.

(Theorem of Jean Dieudonn) Every paracompact Hausdor space is normal. Every paracompact Hausdor space is a shrinking space, that is, every open cover of a paracompact Hausdorspace has a shrinking: another open cover indexed by the same set such that the closure of every set in the newcover lies inside the corresponding set in the old cover.

On paracompact Hausdor spaces, sheaf cohomology and ech cohomology are equal.[5]

8.4. PARACOMPACT HAUSDORFF SPACES 19

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

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

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

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

Proof that paracompact Hausdor spaces admit partitions of unity

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

Lemma 1: If O is a locally nite open cover, then there exists open setsWU for each U 2 O , suchthat each WU U and fWU : U 2 Og is a locally nite renement.

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

PU2O fU is a continuous function which is always non-zero

and nite.

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

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

Now we deneWU =SfA 2 V : A Ug . We have that each WU U ; for otherwise: suppose

there is x 2 WU n U . We will show that there is closed set C WU such that x /2 C (this meanssimply x /2 WU by denition of closure). Since we chose V to be locally nite there is neighbourhoodV [x] of x such that only nitely many sets U1; :::; Un 2 fA 2 V : A Ug have non-empty intersectionwith V [x] . We take their closures U1; :::; Un and then V := V [x] n [ Ui is an open set (since sum isnite) such that V \WU = ? . Moreover x 2 V , because 8i = f1; :::; ng we have Ui U and weknow that x /2 U . Then C := X n V is closed set without x which conatins WU . So x /2 WU andwe've reached contradiction. And it easy to see that fWU : U 2 Og is an open renement of O .

Finally, to verify that this cover is locally nite, x x 2 X and letN be a neighbourhood of x . We knowthat for each U we haveWU U . Since O is locally nite there are only nitely many sets U1; :::; Ukhaving non-empty intersection withN . Then only setsWU1 ; :::;WUk have non-empty intersection withN , because for every other U 0 we have N \WU 0 N \ U 0 = ?


Proof (Lemma 2): Applying Lemma 1, let fU : X ! [0; 1] be coninuous maps with fU WU = 1and supp fU U (by Urysohns lemma for disjoint closed sets in normal spaces, which a paracompactHausdor space is). Note by the support of a function, we here mean the points not mapping to zero(and not the closure of this set). To show that f =PU2O fU is always nite and non-zero, take x 2 X, and let N a neighbourhood of x meeting only nitely many sets in O ; thus x belongs to only nitelymany sets in O ; thus fU (x) = 0 for all but nitely many U ; moreover x 2 WU for some U ,thus fU (x) = 1 ; so f(x) is nite and 1 . To establish continuity, take x;N as before, and letS = fU 2 O : N meets Ug , which is nite; then f N = PU2S fU N , which is a continuousfunction; hence the preimage under f of a neighbourhood of f(x) will be a neighbourhood of x .

Proof (Theorem): TakeO a locally nite subcover of the renement cover: fV open : (9U 2 O) V Ug . Applying Lemma 2, we obtain continuous functions fW : X ! [0; 1] with supp fW W (thusthe usual closed version of the support is contained in some U 2 O , for eachW 2 O ; for which theirsum constitutes a continuous function which is always nite non-zero (hence 1/f is continuous positive,nite-valued). So replacing each fW by fW /f , we have now all things remaining the same thattheir sum is everywhere 1 . Finally for x 2 X , lettingN be a neighbourhood of x meeting only nitelymany sets in O , we have fW N = 0 for all but nitely manyW 2 O since each supp fW W. Thus we have a partition of unity subordinate to the original open cover.

8.5 Relationship with compactnessThere is a similarity between the denitions of compactness and paracompactness: For paracompactness, subcoveris replaced by open renement and nite by is replaced by locally nite. Both of these changes are signicant:if we take the denition of paracompact and change open renement back to subcover, or locally nite back tonite, we end up with the compact spaces in both cases.Paracompactness has little to do with the notion of compactness, but rather more to do with breaking up topologicalspace entities into manageable pieces.

8.5.1 Comparison of properties with compactnessParacompactness is similar to compactness in the following respects:

Every closed subset of a paracompact space is paracompact. Every paracompact Hausdor space is normal.

It is dierent in these respects:

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

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

8.6 VariationsThere are several variations of the notion of paracompactness. To dene them, we rst need to extend the list ofterms above:A topological space is:

metacompact if every open cover has an open pointwise nite renement. orthocompact if every open cover has an open renement such that the intersection of all the open sets aboutany point in this renement is open.

8.7. SEE ALSO 21

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

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

8.6.1 Denition of relevant terms for the variations Given a cover and a point, the star of the point in the cover is the union of all the sets in the cover that containthe point. In symbols, the star of x in U = {U : in A} is

U(x) :=[U3x

U:

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

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

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

f 2 A : x 2 Ug

is nite.

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

8.7 See also a-paracompact space Paranormal space

8.8 Notes[1] Hatcher, Allen, Vector bundles and K-theory, preliminary version available on the authors homepage

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

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

[4] C. Good, I. J. Tree, and W. S. Watson. On Stones Theorem and the Axiom of Choice. Proceedings of the AmericanMathematical Society, Vol. 126, No. 4. (April, 1998), pp. 12111218.

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


8.9 References Dieudonn, Jean (1944), Une gnralisation des espaces compacts, Journal de Mathmatiques Pures et Ap-pliques, Neuvime Srie 23: 6576, ISSN 0021-7824, MR 0013297

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

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

Mathew, Akhil. Topology/Paracompactness.

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

Chapter 9

Regular space

In topology and related elds of mathematics, a topological space X is called a regular space if every non-emptyclosed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods.[1] Thus p and Ccan be separated by neighborhoods. This condition is known as Axiom T3. The term "T3 space" usually means aregular Hausdor space". These conditions are examples of separation axioms.

9.1 Denitions

U

x

V

F

The point x, represented by a dot to the left of the picture, and the closed set F, represented by a closed disk to the right of the picture,are separated by their neighbourhoods U and V, represented by larger open disks. The dot x has plenty of room to wiggle aroundthe open disk U, and the closed disk F has plenty of room to wiggle around the open disk V, yet U and V do not touch each other.

A topological space X is a regular space if, given any nonempty closed set F and any point x that does not belongto F, there exists a neighbourhood U of x and a neighbourhood V of F that are disjoint. Concisely put, it must bepossible to separate x and F with disjoint neighborhoods.A T3 space or regular Hausdor space is a topological space that is both regular and a Hausdor space. (AHausdor space or T2 space is a topological space in which any two distinct points are separated by neighbourhoods.)It turns out that a space is T3 if and only if it is both regular and T0. (A T0 or Kolmogorov space is a topologicalspace in which any two distinct points are topologically distinguishable, i.e., for every pair of distinct points, at leastone of them has an open neighborhood not containing the other.) Indeed, if a space is Hausdor then it is T0, andeach T0 regular space is Hausdor: given two distinct points, at least one of them misses the closure of the other one,so (by regularity) there exist disjoint neighborhoods separating one point from (the closure of) the other.

23

24 CHAPTER 9. REGULAR SPACE

Although the denitions presented here for regular and T3" are not uncommon, there is signicant variation inthe literature: some authors switch the denitions of regular and T3" as they are used here, or use both termsinterchangeably. In this article, we will use the term regular freely, but we will usually say regular Hausdor,which is unambiguous, instead of the less precise T3". For more on this issue, see History of the separation axioms.A locally regular space is a topological space where every point has an open neighbourhood that is regular. Everyregular space is locally regular, but the converse is not true. A classical example of a locally regular space that is notregular is the bug-eyed line.

9.2 Relationships to other separation axioms

A regular space is necessarily also preregular, i.e., any two topologically distinguishable points can be separated byneighbourhoods. Since a Hausdor space is the same as a preregular T0 space, a regular space that is also T0 must beHausdor (and thus T3). In fact, a regular Hausdor space satises the slightly stronger condition T. (However,such a space need not be completely Hausdor.) Thus, the denition of T3 may cite T0, T1, or T instead of T2(Hausdorness); all are equivalent in the context of regular spaces.Speaking more theoretically, the conditions of regularity and T3-ness are related by Kolmogorov quotients. A spaceis regular if and only if its Kolmogorov quotient is T3; and, as mentioned, a space is T3 if and only if its both regularand T0. Thus a regular space encountered in practice can usually be assumed to be T3, by replacing the space withits Kolmogorov quotient.There are many results for topological spaces that hold for both regular and Hausdor spaces. Most of the time, theseresults hold for all preregular spaces; they were listed for regular and Hausdor spaces separately because the ideaof preregular spaces came later. On the other hand, those results that are truly about regularity generally don't alsoapply to nonregular Hausdor spaces.There aremany situations where another condition of topological spaces (such as normality, pseudonormality, paracompactness,or local compactness) will imply regularity if some weaker separation axiom, such as preregularity, is satised. Suchconditions often come in two versions: a regular version and a Hausdor version. Although Hausdor spaces aren'tgenerally regular, a Hausdor space that is also (say) locally compact will be regular, because any Hausdor space ispreregular. Thus from a certain point of view, regularity is not really the issue here, and we could impose a weakercondition instead to get the same result. However, denitions are usually still phrased in terms of regularity, sincethis condition is more well known than any weaker one.Most topological spaces studied in mathematical analysis are regular; in fact, they are usually completely regular,which is a stronger condition. Regular spaces should also be contrasted with normal spaces.

9.3 Examples and nonexamples

A zero-dimensional space with respect to the small inductive dimension has a base consisting of clopen sets. Everysuch space is regular.As described above, any completely regular space is regular, and any T0 space that is not Hausdor (and hence notpreregular) cannot be regular. Most examples of regular and nonregular spaces studied in mathematics may be foundin those two articles. On the other hand, spaces that are regular but not completely regular, or preregular but notregular, are usually constructed only to provide counterexamples to conjectures, showing the boundaries of possibletheorems. Of course, one can easily nd regular spaces that are not T0, and thus not Hausdor, such as an indiscretespace, but these examples provide more insight on the T0 axiom than on regularity. An example of a regular spacethat is not completely regular is the Tychono corkscrew.Most interesting spaces in mathematics that are regular also satisfy some stronger condition. Thus, regular spaces areusually studied to nd properties and theorems, such as the ones below, that are actually applied to completely regularspaces, typically in analysis.There exist Hausdor spaces that are not regular. An example is the set R with the topology generated by sets of theform U C, where U is an open set in the usual sense, and C is any countable subset of U.

9.4. ELEMENTARY PROPERTIES 25

9.4 Elementary propertiesSuppose that X is a regular space. Then, given any point x and neighbourhood G of x, there is a closed neighbourhoodE of x that is a subset of G. In fancier terms, the closed neighbourhoods of x form a local base at x. In fact, thisproperty characterises regular spaces; if the closed neighbourhoods of each point in a topological space form a localbase at that point, then the space must be regular.Taking the interiors of these closed neighbourhoods, we see that the regular open sets form a base for the open setsof the regular space X. This property is actually weaker than regularity; a topological space whose regular open setsform a base is semiregular.

9.5 References[1] Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

Chapter 10

Semiregular space

A semiregular space is a topological space whose regular open sets (sets that equal the interiors of their closures)form a base.Every regular space is semiregular, and every topological space may be embedded into a semiregular space.[1]

Semiregular spaces should not be confused with locally regular spaces, spaces in which there is a base of open setsthat induce regular subspaces. For example, the bug-eyed line is locally regular but not semiregular.

10.1 References[1] Willard, Stephen (2004), 14E. Semiregular spaces, General Topology, Dover, p. 98, ISBN 978-0-486-43479-7.

26

Chapter 11

Separation axiom

For the axiom of set theory, see Axiom schema of separation.In topology and related elds of mathematics, there are several restrictions that one often makes on the kinds of

topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. Theseare sometimes called Tychono separation axioms, after Andrey Tychono.The separation axioms are axioms only in the sense that, when dening the notion of topological space, one couldadd these conditions as extra axioms to get a more restricted notion of what a topological space is. The modernapproach is to x once and for all the axiomatization of topological space and then speak of kinds of topologicalspaces. However, the term separation axiom has stuck. The separation axioms are denoted with the letter T afterthe German Trennungsaxiom, which means separation axiom.The precise meanings of the terms associated with the separation axioms has varied over time, as explained in Historyof the separation axioms. It is important to understand the authors denition of each condition mentioned to knowexactly what they mean, especially when reading older literature.

11.1 Preliminary denitionsBefore we dene the separation axioms themselves, we give concrete meaning to the concept of separated sets (andpoints) in topological spaces. (Separated sets are not the same as separated spaces, dened in the next section.)The separation axioms are about the use of topological means to distinguish disjoint sets and distinct points. Its notenough for elements of a topological space to be distinct (that is, unequal); we may want them to be topologicallydistinguishable. Similarly, its not enough for subsets of a topological space to be disjoint; we may want them to beseparated (in any of various ways). The separation axioms all say, in one way or another, that points or sets that aredistinguishable or separated in some weak sense must also be distinguishable or separated in some stronger sense.Let X be a topological space. Then two points x and y in X are topologically distinguishable if they do not haveexactly the same neighbourhoods (or equivalently the same open neighbourhoods); that is, at least one of them has aneighbourhood that is not a neighbourhood of the other (or equivalently there is an open set that one point belongs tobut the other point does not).Two points x and y are separated if each of them has a neighbourhood that is not a neighbourhood of the other; thatis, neither belongs to the others closure. More generally, two subsets A and B of X are separated if each is disjointfrom the others closure. (The closures themselves do not have to be disjoint.) All of the remaining conditions forseparation of sets may also be applied to points (or to a point and a set) by using singleton sets. Points x and y willbe considered separated, by neighbourhoods, by closed neighbourhoods, by a continuous function, precisely by afunction, if and only if their singleton sets {x} and {y} are separated according to the corresponding criterion.Subsets A and B are separated by neighbourhoods if they have disjoint neighbourhoods. They are separated by closedneighbourhoods if they have disjoint closed neighbourhoods. They are separated by a continuous function if thereexists a continuous function f from the space X to the real lineR such that the image f(A) equals {0} and f(B) equals{1}. Finally, they are precisely separated by a continuous function if there exists a continuous function f from X to Rsuch that the preimage f1({0}) equals A and f1({1}) equals B.These conditions are given in order of increasing strength: Any two topologically distinguishable points must be

27

28 CHAPTER 11. SEPARATION AXIOM

distinct, and any two separated points must be topologically distinguishable. Any two separated sets must be disjoint,any two sets separated by neighbourhoods must be separated, and so on.For more on these conditions (including their use outside the separation axioms), see the articles Separated sets andTopological distinguishability.

11.2 Main denitionsThese denitions all use essentially the preliminary denitions above.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 regularand T3", etc. Many of the concepts also have several names; however, the one listed rst is always least likely to beambiguous.Most of these axioms have alternative denitions with the samemeaning; the denitions given here fall into a consistentpattern that relates the various notions of separation dened in the previous section. Other possible denitions can befound in the individual articles.In all of the following denitions, X is again a topological space.

X is T0, or Kolmogorov, if any two distinct points in X are topologically distinguishable. (It will be a commontheme among the separation axioms to have one version of an axiom that requires T0 and one version thatdoesn't.)

X is R0, or symmetric, if any two topologically distinguishable points in X are separated.

X is T1, or accessible or Frchet, if any two distinct points in X are separated. Thus, X is T1 if and only if itis both T0 and R0. (Although you may say such things as T1 space, Frchet topology, and Suppose thatthe topological space X is Frchet, avoid saying Frchet space in this context, since there is another entirelydierent notion of Frchet space in functional analysis.)

X is R1, or preregular, if any two topologically distinguishable points in X are separated by neighbourhoods.Every R1 space is also R0.

X is Hausdor, or T2 or separated, if any two distinct points in X are separated by neighbourhoods. Thus, Xis Hausdor if and only if it is both T0 and R1. Every Hausdor space is also T1.

X is T2, or Urysohn, if any two distinct points in X are separated by closed neighbourhoods. Every Tspace is also Hausdor.

X is completely Hausdor, or completely T2, if any two distinct points in X are separated by a continuousfunction. Every completely Hausdor space is also T.

X is regular if, given any point x and closed set F in X such that x does not belong to F, they are separated byneighbourhoods. (In fact, in a regular space, any such x and F will also be separated by closed neighbourhoods.)Every regular space is also R1.

X is regular Hausdor, or T3, if it is both T0 and regular.[1] Every regular Hausdor space is also T.

X is completely regular if, given any point x and closed set F in X such that x does not belong to F, they areseparated by a continuous function. Every completely regular space is also regular.

X is Tychono, or T3, completely T3, or completely regular Hausdor, if it is both T0 and completelyregular.[2] Every Tychono space is both regular Hausdor and completely Hausdor.

X is normal if any two disjoint closed subsets of X are separated by neighbourhoods. (In fact, a space is normalif and only if any two disjoint closed sets can be separated by a continuous function; this is Urysohns lemma.)

11.3. RELATIONSHIPS BETWEEN THE AXIOMS 29

X is normal Hausdor, or T4, if it is both T1 and normal. Every normal Hausdor space is both Tychonoand normal regular.

X is completely normal if any two separated sets are separated by neighbourhoods. Every completely normalspace is also normal.

X is completely normal Hausdor, or T5 or completely T4, if it is both completely normal and T1. Everycompletely normal Hausdor space is also normal Hausdor.

X is perfectly normal if any two disjoint closed sets are precisely separated by a continuous function. Everyperfectly normal space is also completely normal.

X is perfectly normal Hausdor, or T6 or perfectly T4, if it is both perfectly normal and T1. Every perfectlynormal Hausdor space is also completely normal Hausdor.

11.3 Relationships between the axiomsThe T0 axiom is special in that it can be not only added to a property (so that completely regular plus T0 is Tychono)but also subtracted from a property (so that Hausdor minus T0 is R1), in a fairly precise sense; see Kolmogorovquotient for more information. When applied to the separation axioms, this leads to the relationships in the tablebelow:In this table, you go from the right side to the left side by adding the requirement of T0, and you go from the left sideto the right side by removing that requirement, using the Kolmogorov quotient operation. (The names in parenthesesgiven on the left side of this table are generally ambiguous or at least less well known; but they are used in the diagrambelow.)Other than the inclusion or exclusion of T0, the relationships between the separation axioms are indicated in thefollowing diagram:In this diagram, the non-T0 version of a condition is on the left side of the slash, and the T0 version is on the rightside. Letters are used for abbreviation as follows: P = perfectly, C = completely, N = normal, and R(without a subscript) = regular. A bullet indicates that there is no special name for a space at that spot. The dash atthe bottom indicates no condition.You can combine two properties using this diagram by following the diagram upwards until both branches meet.For example, if a space is both completely normal (CN) and completely Hausdor (CT2"), then following bothbranches up, you nd the spot "/T5". Since completely Hausdor spaces are T0 (even though completely normalspaces may not be), you take the T0 side of the slash, so a completely normal completely Hausdor space is the sameas a T5 space (less ambiguously known as a completely normal Hausdor space, as you can see in the table above).As you can see from the diagram, normal and R0 together imply a host of other properties, since combining the twoproperties leads you to follow a path through the many nodes on the rightside branch. Since regularity is the mostwell known of these, spaces that are both normal and R0 are typically called normal regular spaces. In a somewhatsimilar fashion, spaces that are both normal and T1 are often called normal Hausdor spaces by people that wishto avoid the ambiguous T notation. These conventions can be generalised to other regular spaces and Hausdorspaces.

11.4 Other separation axiomsThere are some other conditions on topological spaces that are sometimes classied with the separation axioms, butthese don't t in with the usual separation axioms as completely. Other than their denitions, they aren't discussedhere; see their individual articles.

X is semiregular if the regular open sets form a base for the open sets of X. Any regular space must also besemiregular.


X is quasi-regular if for any nonempty open set G, there is a nonempty open set H such that the closure of His contained in G.

X is fully normal if every open cover has an open star renement. X is fully T4, or fully normal Hausdor,if it is both T1 and fully normal. Every fully normal space is normal and every fully T4 space is T4. Moreover,one can show that every fully T4 space is paracompact. In fact, fully normal spaces actually have more to dowith paracompactness than with the usual separation axioms.

X is sober if, for every closed set C that is not the (possibly nondisjoint) union of two smaller closed sets, thereis a unique point p such that the closure of {p} equals C. More briey, every irreducible closed set has a uniquegeneric point. Any Hausdor space must be sober, and any sober space must be T0.

11.5 See also General topology

11.6 Sources Schechter, Eric (1997). Handbook of Analysis and its Foundations. San Diego: Academic Press. ISBN0126227608. (has Ri axioms, among others)

Willard, Stephen (1970). General topology. Reading, Mass.: Addison-Wesley Pub. Co. ISBN 0-486-43479-6.(has all of the non-Ri axioms mentioned in the Main Denitions, with these denitions)

Merrield, Richard E.; Simmons, Howard E. (1989). Topological Methods in Chemistry. New York: Wiley.ISBN 0-471-83817-9. (gives a readable introduction to the separation axioms with an emphasis on nitespaces)

[1] Schechter, p. 441

[2] Schechter, p. 443

11.7 External links Separation Axioms at ProvenMath Table of separation and metrisability axioms from Schechter

11.7. EXTERNAL LINKS 31

An illustration of some of the separation axioms. A blue region indicates an open set, a red rectangle a closed set, and a black dot apoint.


Hasse diagram of the separation axioms.

Chapter 12

Sober space

In mathematics, a sober space is a topological space such that every irreducible closed subset of X is the closure ofexactly one point of X: that is, this closed subset has a unique generic point.

12.1 Properties and examples

Any Hausdor (T2) space is sober (the only irreducible subsets being points), and all sober spaces are Kolmogorov(T0), and both implications are strict.[1] Sobriety is not comparable to the T1 condition: an example of a T1 sp

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Separation Axioms