# Definition:T5 Space/Definition 2

## Definition

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

$\struct {S, \tau}$ is a **$T_5$ space** if and only if:

- $\forall Y,A \subseteq S: (A \subseteq Y^\circ \wedge A^- \subseteq Y) \implies \exists N \subseteq Y: \relcomp S N \in \tau: \exists U \in \tau: A \subseteq U \subseteq N$

That is:

- $\struct {S, \tau}$ is a
**$T_5$ space**if and only if every subset $Y \subseteq S$ contains a closed neighborhood of each $A \subseteq Y^\circ$ for which $A^- \subseteq Y$.

In the above, $Y^\circ$ denotes the interior of $Y$ and $A^-$ denotes the closure of $A$.

## Variants of Name

From about 1970, treatments of this subject started to refer to this as a **completely normal space**, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a completely normal space as a **$T_5$ 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
**$T_5$ 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