Completely regular space

This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces

In the T family (properties of topological spaces related to separation axioms), this is called: T3.5

This article is about a basic definition in topology.
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Definition

A topological space is termed completely regular if it satisfies the following equivalent conditions:

• Given any point and any closed subset, there is a continuous function on the topological space that takes the value $0$ at the point and $1$ at the closed subset
• It occurs as the underlying topological space of a uniform space
• It possesses a compactification

Relation with other properties

Stronger properties

• Normal space
• Underlying space of a topological group