Definition:Regular Space

Definition
Let $T = \left({X, \vartheta}\right)$ be a topological space.

$\left({X, \vartheta}\right)$ is a regular space iff:


 * $\forall F \in \complement \left({\vartheta}\right), y \in \complement_X \left({F}\right): \exists U, V \in \vartheta: F \subseteq U, y \in V: U \cap V = \varnothing$

That is, for any closed set $F \subseteq X$ and any point $y \in X$ such that $y \notin F$ there exist disjoint open sets $U, V \in \vartheta$ such that $F \subseteq U$, $y \in V$.

That is:
 * $\left({X, \vartheta}\right)$ is regular when any closed set $F \subseteq X$ and any point not in $F$ are separated by neighborhoods.

Equivalent Definitions

 * $\left({X, \vartheta}\right)$ is regular iff each open set contains a closed neighborhood around each of its points.


 * $\left({X, \vartheta}\right)$ is regular iff each of its closed sets is the intersection of its closed neighborhoods.

This is proved in Equivalent Definitions for Regular Space.

Variants of Name
Earlier (pre-1970) treatment of this subject tends to refer to this as a $T_3$ space, and what we define as a $T_3$ space as a regular space.