# Definition:Completely Hausdorff Space

## Definition

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

$\left({S, \tau}\right)$ is a completely Hausdorff space or $T_{2 \frac 1 2}$ space if and only if:

$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U^- \cap V^- = \varnothing$

That is, for any two distinct elements $x, y \in S$ there exist open sets $U, V \in \tau$ containing $x$ and $y$ respectively whose closures are disjoint.

That is:

$\left({S, \tau}\right)$ is a $T_{2 \frac 1 2}$ space if and only if every two points in $S$ are separated by closed neighborhoods.

## Source of Name

This entry was named for Felix Hausdorff.

## Variants of Name

From about 1970, treatments of this subject started to refer to this as an Urysohn space, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as an Urysohn space as a completely Hausdorff 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 completely Hausdorff spaces can be found here.