Urysohn space

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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