# 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.

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 2$: Regular and Normal Spaces - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**T-axioms**or**Tychonoff conditions**:**4.**(**$T_4$ space**)