Definition:T3 Space

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

$T = \left({X, \vartheta}\right)$ is a $T_3$ space iff:


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

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 $T_3$ when any closed set $F \subseteq X$ and any point not in $F$ are separated by neighborhoods.

Equivalent Definitions

 * $T = \left({X, \vartheta}\right)$ is $T_3$ iff each open set contains a closed neighborhood around each of its points.


 * $T = \left({X, \vartheta}\right)$ is $T_3$ iff each of its closed sets is the intersection of its closed neighborhoods.

This is proved in Equivalent Definitions for $T_3$ Space.