Editing Normal space (section)

== 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 '''T<sub>4</sub> space''' is a [[T1 space|T<sub>1</sub> 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 '''T<sub>5</sub> space''', or '''completely T<sub>4</sub> space''', is a completely normal T<sub>1</sub> space ''X'', which implies that ''X'' is Hausdorff; equivalently, every subspace of ''X'' must be a T<sub>4</sub> 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|G<sub>δ</sub> 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 T<sub>1</sub> 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 '''T<sub>6</sub> space''', or '''perfectly T<sub>4</sub> space''', is a perfectly normal Hausdorff space.

Note that the terms "normal space" and "T<sub>4</sub>" and derived concepts occasionally have a different meaning. (Nonetheless, "T<sub>5</sub>" always means the same as "completely T<sub>4</sub>", whatever the meaning of T<sub>4</sub> 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 T<sub>4</sub> space. Given the historical confusion of the meaning of the terms, verbal descriptions when applicable are helpful, that is, "normal Hausdorff" instead of "T<sub>4</sub>", or "completely normal Hausdorff" instead of "T<sub>5</sub>".

[[Fully normal space]]s and [[paracompact Hausdorff space|fully T<sub>4</sub> 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]].