# 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.