# Definition:Urysohn Space

## Definition

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

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

- For any distinct elements $x, y \in S$ (i.e. $x \ne y$), there exists an Urysohn function for $\left\{{x}\right\}$ and $\left\{{y}\right\}$.

## Source of Name

This entry was named for Pavel Samuilovich Urysohn.

## Variants of Name

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

## Sources

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