# Definition:Urysohn Space

## Definition

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

$\struct {S, \tau}$ is an **Urysohn space** if and only if:

- For any distinct elements $x, y \in S$ (that is, $x \ne y$), there exists an Urysohn function for $\set x$ and $\set y$.

## 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 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 see

- Results about
**Urysohn spaces**can be found here.

## 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: Additional Separation Properties