# Definition:T5 Space

## Definition

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

### Definition 1

$\left({S, \tau}\right)$ is a **$T_5$ space** if and only if:

- $\forall A, B \subseteq S, A^- \cap B = A \cap B^- = \varnothing: \exists U, V \in \tau: A \subseteq U, B \subseteq V, U \cap V = \varnothing$

That is:

- $\left({S, \tau}\right)$ is a
**$T_5$ space**when for any two separated sets $A, B \subseteq S$ there exist disjoint open sets $U, V \in \tau$ containing $A$ and $B$ respectively.

### Definition 2

$\left({S, \tau}\right)$ 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 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.

## Also see

- Results about
**$T_5$ spaces**can be found here.