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Examples of normal spaces

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{{Short description|Type of topological space}}
{{for|normal vector space|normal (geometry)}}
{{Separation axioms}}

In [[topology]] and related branches of [[mathematics]], a '''normal space''' is a [[topological space]] in which any two disjoint [[closed set]]s have disjoint [[open neighborhood]]s. Such spaces need not be [[Hausdorff space|Hausdorff]] in general. A normal Hausdorff space is called a '''T4 space'''. Strengthenings of these concepts are detailed in the article below and include '''completely normal spaces''' and '''perfectly normal spaces''', and their Hausdorff variants: '''T5 spaces''' and '''T6 spaces'''.
All these conditions are examples of [[separation axiom]]s.

== Definitions ==

A [[topological space]] ''X'' is a '''normal space''' if, given any [[disjoint sets|disjoint]] [[closed set]]s ''E'' and ''F'', there are [[neighbourhood (topology)|neighbourhoods]] ''U'' of ''E'' and ''V'' of ''F'' that are also disjoint. More intuitively, this condition says that ''E'' and ''F'' can be [[separated set|separated by neighbourhoods]].

[[File:Normal space.svg|thumb|203px|The closed sets ''E'' and ''F'', here represented by closed disks on opposite sides of the picture, are separated by their respective neighbourhoods ''U'' and ''V'', here represented by larger, but still disjoint, open disks.]]

A '''T4 space''' is a [[T1 space|T1 space]] ''X'' that is normal; this is equivalent to ''X'' being normal and [[Hausdorff space|Hausdorff]].

A '''completely normal space''', or '''{{visible anchor|hereditarily normal space}}''', is a topological space ''X'' such that every [[subspace (topology)|subspace]] of ''X'' is a normal space. It turns out that ''X'' is completely normal if and only if every two [[separated set]]s can be separated by neighbourhoods. Also, ''X'' is completely normal if and only if every open subset of ''X'' is normal with the subspace topology.

A '''T5 space''', or '''completely T4 space''', is a completely normal T1 space ''X'', which implies that ''X'' is Hausdorff; equivalently, every subspace of ''X'' must be a T4 space.

A '''perfectly normal space''' is a topological space <math>X</math> in which every two disjoint closed sets <math>E</math> and <math>F</math> can be [[precisely separated by a function]], in the sense that there is a continuous function <math>f</math> from <math>X</math> to the interval <math>[0,1]</math> such that <math>f^{-1}(\{0\})=E</math> and <math>f^{-1}(\{1\})=F</math>.<ref>Willard, Exercise 15C</ref> This is a stronger separation property than normality, as by [[Urysohn's lemma]] disjoint closed sets in a normal space can be [[separated by a function]], in the sense of <math>E\subseteq f^{-1}(\{0\})</math> and <math>F\subseteq f^{-1}(\{1\})</math>, but not precisely separated in general. It turns out that ''X'' is perfectly normal if and only if ''X'' is normal and every closed set is a [[G-delta set|Gδ set]]. Equivalently, ''X'' is perfectly normal if and only if every closed set is the [[zero set]] of a [[continuous function]]. The equivalence between these three characterizations is called '''Vedenissoff's theorem'''.<ref>Engelking, Theorem 1.5.19. This is stated under the assumption of a T1 space, but the proof does not make use of that assumption.</ref><ref>{{cite web |title=Why are these two definitions of a perfectly normal space equivalent? |url=https://math.stackexchange.com/questions/72138}}</ref> Every perfectly normal space is completely normal, because perfect normality is a [[hereditary property]].<ref>Engelking, Theorem 2.1.6, p. 68</ref><ref name="Munkres p213">{{harvnb|Munkres|2000|p=213}}</ref>

A '''T6 space''', or '''perfectly T4 space''', is a perfectly normal Hausdorff space.

Note that the terms "normal space" and "T4" and derived concepts occasionally have a different meaning. (Nonetheless, "T5" always means the same as "completely T4", whatever the meaning of T4 may be.) The definitions given here are the ones usually used today. For more on this issue, see [[History of the separation axioms]].

Terms like "normal [[regular space]]" and "normal Hausdorff space" also turn up in the literature—they simply mean that the space both is normal and satisfies the other condition mentioned. In particular, a normal Hausdorff space is the same thing as a T4 space. Given the historical confusion of the meaning of the terms, verbal descriptions when applicable are helpful, that is, "normal Hausdorff" instead of "T4", or "completely normal Hausdorff" instead of "T5".

[[Fully normal space]]s and [[paracompact Hausdorff space|fully T4 space]]s 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. Every normal space is locally normal, but the converse is not true. A classical example of a completely regular locally normal space that is not normal is the [[Nemytskii plane]].

== Examples of normal spaces ==
Most spaces encountered in [[mathematical analysis]] are normal Hausdorff spaces, or at least normal regular spaces:
* All [[metric spaces]] (and hence all [[metrizable space]]s) are perfectly normal Hausdorff;
* All [[pseudometric space]]s (and hence all [[pseudometrizable space]]s) are perfectly normal regular, although not in general Hausdorff;
* All [[compact space|compact]] Hausdorff spaces are normal;
* In particular, the [[Stone–Čech compactification]] of a [[Tychonoff space]] is normal Hausdorff;
* Generalizing the above examples, all [[paracompact]] Hausdorff spaces are normal, and all paracompact regular spaces are normal;
* All paracompact [[topological manifold]]s are perfectly normal Hausdorff. However, there exist non-paracompact manifolds that are not even normal.
* All [[order topology|order topologies]] on [[totally ordered set]]s are hereditarily normal and Hausdorff.
* Every regular [[second-countable space]] is completely normal, and every regular [[Lindelöf space]] is normal.

Also, all [[fully normal space]]s are normal (even if not regular). [[Sierpiński space]] is an example of a normal space that is not regular.

== Examples of non-normal spaces ==
An important example of a non-normal topology is given by the [[Zariski topology]] on an [[algebraic variety]] or on the [[spectrum 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 [[function (mathematics)|function]]s from the [[real line]] '''R''' to itself, with the [[topology of pointwise convergence]].
More generally, a theorem of [[Arthur Harold Stone]] states that the [[product topology|product]] of [[uncountable|uncountably many]] non-[[compact space|compact]] metric spaces is never normal.

== Properties ==
Every closed subset of a normal space is normal. The continuous and closed image of a normal space is normal.{{sfn|Willard|1970|pp=[https://archive.org/details/generaltopology00will_0/page/100 100–101]}}

The main significance of normal spaces lies in the fact that they admit "enough" [[continuous function (topology)|continuous]] [[real number|real]]-valued [[function (mathematics)|function]]s, as expressed by the following theorems valid for any normal space ''X''.

[[Urysohn's lemma]]:
If ''A'' and ''B'' are two [[Disjoint sets|disjoint]] closed subsets of ''X'', then there exists a continuous function ''f'' from ''X'' to 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 be entirely 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'' to '''R''', then there exists a continuous function ''F'': ''X'' → '''R''' that extends ''f'' in the sense that ''F''(''x'') = ''f''(''x'') for all ''x'' in ''A''.

The map ''<math>\emptyset\rightarrow X</math>'' has the [[lifting property]] with respect to a map from a certain finite topological space with five points (two open and three closed) to the space with one open and two closed points.<ref>{{Cite web|url=https://ncatlab.org/nlab/show/separation+axioms##TableOfMainSeparationAxiomsAsLiftingProperties|title=separation axioms in nLab|website=ncatlab.org|access-date=2021-10-12}}</ref>

If '''U''' is a locally finite [[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 satisfies any one of these three conditions must be normal.

A [[product space|product]] of normal spaces is not necessarily normal. This fact was first proved by [[Robert Sorgenfrey]]. An example of this phenomenon is the [[Sorgenfrey plane]]. In fact, since there exist spaces which are [[Dowker space|Dowker]], a product of a normal space and [0, 1] need not to be normal. Also, a subset of a normal space need not be normal (i.e. not every normal Hausdorff space is a completely normal Hausdorff space), since every Tychonoff space is a subset of its Stone–Čech compactification (which is normal Hausdorff). A more explicit example is the [[Tychonoff plank]]. The only large class of product spaces of normal spaces known to be normal are the products of compact Hausdorff spaces, since both compactness ([[Tychonoff's theorem]]) and the T2 axiom are preserved under arbitrary products.{{sfn|Willard|1970|loc=Section 17}}

== Relationships to other separation axioms ==
If a normal space is [[R0 space|R0]], then it is in fact [[completely regular]].
Thus, anything from "normal R0" to "normal completely regular" is the same as what we usually call ''normal regular''.
Taking [[Kolmogorov quotient]]s, we see that all normal [[T1 space|T1 space]]s are [[Tychonoff space|Tychonoff]].
These are what we usually call ''normal Hausdorff'' spaces.

A topological space is said to be [[pseudonormal space|pseudonormal]] if given two disjoint closed sets in it, one of which is countable, there are 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.
Specifically, [[Sierpiński space]] is normal but not regular, while the space of functions from '''R''' to itself is Tychonoff but not normal.

==See also==

* {{annotated link|Collectionwise normal space}}
* {{annotated link|Monotonically normal space}}

== Citations ==
{{reflist}}

==References==

*[[Ryszard Engelking|Engelking, Ryszard]], ''General Topology'', Heldermann Verlag Berlin, 1989. {{ISBN|3-88538-006-4}}
*{{cite encyclopedia |last= Kemoto |first= Nobuyuki |editor= K.P. Hart |editor2=J. Nagata |editor3=J.E. Vaughan |title= Higher Separation Axioms |encyclopedia= Encyclopedia of General Topology |publisher= [[Elsevier Science]] |location= Amsterdam |year= 2004 |isbn=978-0-444-50355-8}}
*{{cite book |last=Munkres |first=James R. |author-link=James Munkres |title=Topology |year=2000 |edition=2nd |publisher=[[Prentice-Hall]] |isbn=978-0-13-181629-9}}
*{{cite journal |last= Sorgenfrey |first= R.H.|year= 1947 |title= On the topological product of paracompact spaces |journal= Bull. Amer. Math. Soc. |volume= 53 |issue= 6 |pages= 631–632 |doi=10.1090/S0002-9904-1947-08858-3 |doi-access= free }}
*{{cite journal |last= Stone |first= A. H. |year= 1948 |title=Paracompactness and product spaces |journal=Bull. Amer. Math. Soc. |volume=54 |issue= 10 |pages= 977–982 |doi=10.1090/S0002-9904-1948-09118-2 |doi-access= free}}
*{{cite book |last= Willard |first= Stephen |title= General Topology |publisher= Addison-Wesley |location= Reading, MA |year= 1970 |isbn= 978-0-486-43479-7 |url= https://archive.org/details/generaltopology00will_0 |url-access= registration}}

[[Category:Properties of topological spaces]]
[[Category:Separation axioms]]

Normal space - Revision history

imported>Axoluna: /* Examples of normal spaces */