A topological space records which subsets count as open, but many arguments in analysis need more than open sets. They need the ability to separate two incompatible pieces of information by disjoint neighbourhoods, and later by a [continuous function](/page/Continuous%20Function). Normality is the separation axiom that makes this possible for closed sets.

The point is not cosmetic. If two closed sets cannot be pulled apart, then a function that should be zero on one set and one on the other may not exist. Partitions of unity, extension theorems, and many compactness arguments depend on the same local-to-global manoeuvre: isolate closed conditions without disturbing the rest of the space.

[example: Separating Two Closed Intervals in $\mathbb{R}$] Let $X=\mathbb{R}$ with its usual topology, and let $A=[0,1]$ and $B=[3,4]$. The sets $A$ and $B$ are closed intervals, hence closed in the usual topology, and they are disjoint because every $a\in A$ satisfies $a\leq 1$ while every $b\in B$ satisfies $b\geq 3$. The [open set](/page/Open%20Set) $U=(-1,2)$ contains $A$, since $0>-1$ and $1<2$. The open set $V=(2,5)$ contains $B$, since $3>2$ and $4<5$. Also $U\cap V=\varnothing$: if $x\in U\cap V$, then $x<2$ from $x\in U$ and $x>2$ from $x\in V$, which is impossible. The set-to-set distance is exactly $2$. For any $a\in[0,1]$ and $b\in[3,4]$, we have $a\leq 1<3\leq b$, so $|a-b|=b-a$. Since $b\geq 3$ and $a\leq 1$, \begin{align*} |a-b|=b-a\geq 3-1=2. \end{align*} Therefore $2$ is a lower bound for $\{|a-b|:a\in A,\ b\in B\}$. Taking $a=1$ and $b=3$ gives \begin{align*} |1-3|=2, \end{align*} so the lower bound is attained, and hence \begin{align*} \operatorname{dist}(A,B)=\inf\{|a-b|:a\in A,\ b\in B\}=2. \end{align*} This positive gap is what the open buffers $(-1,2)$ and $(2,5)$ are exploiting: each closed interval is surrounded by an open neighbourhood, and the two neighbourhoods still do not meet. [/example]

In metric spaces this behaviour is so familiar that it can become invisible. Normal spaces isolate exactly the topological content of that separation phenomenon, without mentioning distances.

## Definition

A topological space already tells us what open and closed sets are. Normality asks for an additional promise: whenever two closed conditions are mutually exclusive, the topology has enough open sets to keep them apart.

[definition: Normal Topological Space] A topological space $(X,\tau)$ is a normal topological space if for every pair of closed sets $A,B \subset X$ with $A \cap B = \varnothing$, there exist open sets $U,V \in \tau$ such that \begin{align*} A \subset U, \quad B \subset V, \quad U \cap V = \varnothing. \end{align*} [/definition]

This definition is a child of the general notion of a [topological space](/page/Topological%20Space): the underlying object is still just $(X,\tau)$, but the topology is required to have a stronger separation property. The condition is about closed sets, not arbitrary subsets, because closed sets encode constraints that are stable under limits.

## Separation Axioms and Neighbourhoods

Normality is easiest to use once we separate the page-topic definition from the auxiliary language around it. The first auxiliary notion names the open sets that surround a set, since normality is a statement about choosing two such surrounding sets with no overlap.

[definition: Open Neighbourhood of a Set] Let $(X,\tau)$ be a topological space and let $A \subset X$. An open neighbourhood of $A$ is an open set $U \in \tau$ such that $A \subset U$. [/definition]

With this terminology, normality says that disjoint closed sets have disjoint open neighbourhoods. The definition also explains why normality strengthens the Hausdorff intuition: points are small closed sets in many spaces, but closed sets may be much larger and harder to separate. To turn that intuition into a theorem, we must know when points themselves are closed enough for normality to apply.

[definition: $T_1$ Topological Space] A topological space $(X,\tau)$ is a $T_1$ topological space if for every pair of distinct points $x,y \in X$, there exists an open set $U \in \tau$ such that $x \in U$ and $y \notin U$. [/definition]

The next question is whether closed-set separation recovers the usual separation of distinct points. In a $T_1$ space, singleton sets are closed, so normality can be applied to $\{x\}$ and $\{y\}$. This gives the first structural consequence of the definition and places normality among the standard separation axioms.

[quotetheorem:9166]

This theorem places normality among the separation axioms. It is stronger than Hausdorffness under the usual $T_1$ convention, because it separates all disjoint closed sets rather than only two singleton sets.

[remark: Convention on Normality] Some authors build the $T_1$ condition into the phrase "normal space." On this page, normality means only closed-set separation, and the $T_1$ hypothesis is stated separately whenever it is needed. [/remark]

## Metric Sources of Normality

The most reliable supply of normal spaces comes from metric spaces. A metric gives a numerical way to measure how far a point is from a [closed set](/page/Closed%20Set), and these distance functions turn closed-set separation into inequalities.

To use this mechanism, we need the distance from a point to a set. The definition is metric rather than purely topological, and it is exactly the device that later disappears from the final topological statement.