Definition:T3 Space/Definition 3
Definition
Let $T = \struct {S, \tau}$ be a topological space.
$T = \struct {S, \tau}$ is a $T_3$ space if and only if each of its closed sets is the intersection of its closed neighborhoods:
- $\forall H \subseteq S: \relcomp S H \in \tau: H = \bigcap \set {N_H: \relcomp S {N_H} \in \tau, \exists V \in \tau: H \subseteq V \subseteq N_H}$
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