Functionally Hausdorff space

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

This is a variation of Hausdorffness. View other variations of Hausdorffness

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

Definition

A topological space is termed a completely Hausdorff space or Urysohn space if for any two points in it, there is a continuous function from the whole space to $[0,1]$ that takes the value $0$ at one point and $1$ at the other.

Relation with other properties

Stronger properties

• Normal space
• Completely regular space

Weaker properties

• Hausdorff space

Facts

Any connected Urysohn space with at least two points is uncountable (more precisely, its cardinality must be at least that of the continuum). This follows from the fact that its image under any continuous function must be connected, and hence the Urysohn function separating two points must be surjective to $[0,1]$. For full proof, refer: Connected Urysohn implies uncountable