# Definition:T3 1/2 Space

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## 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.

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms: Completely Regular Spaces