Definition:Regular Space

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

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

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


 * $\forall x, y \in X$, either:
 * $\exists U \in \vartheta: x \in U, y \notin U$
 * $\exists U \in \vartheta: y \in U, x \notin U$