# Urysohn space

## Definition

A topological space $X$ is termed a Urysohn space if, for any two distinct points $x,y \in X$, there exist disjoint open subsets $U \ni x, V \ni y$ such that the closures <math\overline{U}[/itex] and $\overline{V}$ are disjoint closed subsets of $X$.

Note that the term Urysohn space is also used for the somewhat stronger notion of functionally Hausdorff space. There is a terminological ambiguity here.

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

In the T family (properties of topological spaces related to separation axioms), this is called: T2.5

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
regular Hausdorff space (also called $T_3$) T1 and any point can be separated from any disjoint closed subset regular Hausdorff implies Urysohn Urysohn not implies regular |FULL LIST, MORE INFO
Tychonoff space (also called $T_{3.5}$) T1 and any point and disjoint closed subset can be separated by a continuous function Functionally Hausdorff space|FULL LIST, MORE INFO
normal Hausdorff space (also called $T_4$) T1 and normal Functionally Hausdorff space|FULL LIST, MORE INFO