Definition:Regular Space
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Definition
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.
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: Regular and Normal Spaces
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): T-axioms or Tychonoff conditions: 3a. ($T_3$ space)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): regular space
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): regular space