# Urysohn space

From Topospaces

## Contents

## Definition

A topological space is termed a **Urysohn space** if, for any two *distinct* points , there exist disjoint open subsets such that the closures <math\overline{U}</math> and are disjoint closed subsets of .

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 ) | 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 ) | 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 ) | 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 |