Definition:Regular Space

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

$\struct {S, \tau}$ is a regular space if and only if:

$\struct {S, \tau}$ is a $T_3$ space
$\struct {S, \tau}$ is a $T_0$ (Kolmogorov) space.

That is:

$\forall F \subseteq S: \relcomp S F \in \tau, y \in \relcomp S F: \exists U, V \in \tau: F \subseteq U, y \in V: U \cap V = \O$
$\forall x, y \in S$, either:
$\exists U \in \tau: x \in U, y \notin U$
$\exists U \in \tau: y \in U, x \notin U$

Variants of Name

From about 1970, treatments of this subject started to refer to this as a $T_3$ space, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a $T_3$ space as a regular 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 regular spaces can be found here.