Definition:T3 Space/Definition 2
Definition
Let $T = \struct {S, \tau}$ be a topological space.
$T = \struct {S, \tau}$ is $T_3$ if and only if each open set contains a closed neighborhood around each of its points:
- $\forall U \in \tau: \forall x \in U: \exists N_x: \relcomp S {N_x} \in \tau: \exists V \in \tau: x \in V \subseteq N_x \subseteq U$
where $N_x$ denotes a neighborhood of $x$.
Variants of Name
From about 1970, treatments of this subject started to refer to this as a regular space, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a regular space as a $T_3$ 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 $T_3$ 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