# Definition:T4 Space/Definition 2

## Definition

Let $T = \struct {S, \tau}$ be a topological space.

$T = \struct {S, \tau}$ is **$T_4$** if and only if each open set $U$ contains a closed neighborhood of each closed set contained in $U$.

## Variants of Name

From about 1970, treatments of this subject started to refer to this as a **normal space**, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a normal space as a **$T_4$ 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 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: *Counterexamples in Topology* (2nd ed.).

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

## Also see

- Results about
**$T_4$ spaces**can be found here.

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms