# Definition:T3 1/2 Space

## Definition

Let $T = \struct {S, \tau}$ be a topological space.

$\struct {S, \tau}$ is a $T_{3 \frac 1 2}$ space if and only if:

For any closed set $F \subseteq S$ and any point $y \in S$ such that $y \notin F$, there exists an Urysohn function for $F$ and $\set y$.

## Variants of Name

From about 1970, treatments of this subject started to refer to this as a completely regular space, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a completely regular space as a $T_{3 \frac 1 2}$ space.

However, the names are to a fair extent arbitrary and a matter of taste, as there appears to be no completely satisfactory system for naming all these various Tychonoff separation axioms.

The system as used here broadly follows 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.).

The system used on the Separation axiom page at Wikipedia differs from this.

## Also see

• Results about $T_{3 \frac 1 2}$ spaces can be found here.