# Definition:Normal Space

## Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

$\left({S, \tau}\right)$ is a normal space if and only if:

$\left({S, \tau}\right)$ is a $T_4$ space
$\left({S, \tau}\right)$ is a $T_1$ (Fréchet) space.

That is:

$\forall A, B \in \complement \left({\tau}\right), A \cap B = \varnothing: \exists U, V \in \tau: A \subseteq U, B \subseteq V, U \cap V = \varnothing$
$\forall x, y \in S$, both:
$\exists U \in \tau: x \in U, y \notin U$
$\exists V \in \tau: y \in V, x \notin V$

## Variants of Name

From about 1970, treatments of this subject started to refer to this as a $T_4$ space, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a $T_4$ space as a normal space.

However, the names are to a fair extent arbitrary and a matter of taste, as there appears to be no completely satisfactory system for naming all these various Tychonoff separation axioms.

The system as used here broadly follows 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology.

The system used on the Separation axiom page at Wikipedia differs from this.

This space is also referred to as normal Hausdorff.

## Also see

• Results about normal spaces can be found here.