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}</math> 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
functionally Hausdorff space				|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

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Hausdorff space				|FULL LIST, MORE INFO
T1 space				Hausdorff space|FULL LIST, MORE INFO
Kolmogorov space				Hausdorff space|FULL LIST, MORE INFO