Definition:Regular Space

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

$\left({S, \tau}\right)$ is a regular space iff:
 * $\left({S, \tau}\right)$ is a $T_3$ space
 * $\left({S, \tau}\right)$ is a $T_0$ (Kolmogorov) space.

That is:
 * $\forall F \subseteq S: \complement_S \left({F}\right) \in \tau, y \in \complement_S \left({F}\right): \exists U, V \in \tau: F \subseteq U, y \in V: U \cap V = \varnothing$


 * $\forall x, y \in S$, either:
 * $\exists U \in \tau: x \in U, y \notin U$
 * $\exists U \in \tau: y \in U, x \notin U$