# Definition:Completely Hausdorff Space

## Definition

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

### Definition 1

$\struct {S, \tau}$ is a **completely Hausdorff space** or **$T_{2 \frac 1 2}$ space** if and only if:

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

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.

### Definition 2

$\struct {S, \tau}$ is a **completely Hausdorff space** or **$T_{2 \frac 1 2}$ space** if and only if:

- $\forall x, y \in S, x \ne y : \exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x^- \cap N_y^- = \O$

That is:

- $\struct {S, \tau}$ is a
**$T_{2 \frac 1 2}$ space**if and only if every two points in $S$ are separated by closed neighborhoods.

## 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 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 known as

Some sources give this as **$T_{\frac 5 2}$ space**, which of course evaluates to the same as a **$T_{2 \frac 1 2}$ space**.

## Also see

- Results about
**completely Hausdorff spaces**can be found here.

## Source of Name

This entry was named for Felix Hausdorff.

## 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: Completely Hausdorff Spaces - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**T-axioms**or**Tychonoff conditions**:**2b.**(**$T_{\frac 5 2}$ space**or**Urysohn space**)