# 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.

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 2$: Completely Hausdorff Spaces