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