Definition:T3 Space

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

$\left({X, \vartheta}\right)$ is a $T_3$ space iff:
 * $\left({X, \vartheta}\right)$ is a regular space
 * $\left({X, \vartheta}\right)$ is a Kolmogorov ($T_0$) 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:
 * $\exists U \in \vartheta: x \in U, y \notin U$
 * $\exists U \in \vartheta: y \in U, x \notin U$

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