Editing Normal space (section)

== 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 T<sub>2</sub> axiom are preserved under arbitrary products.{{sfn|Willard|1970|loc=Section 17}}