Definition:T3 Space

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

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


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

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

That is:
 * $\left({S, \tau}\right)$ is $T_3$ when any closed set $F \subseteq S$ and any point not in $F$ are separated by neighborhoods.

Equivalent Definitions

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


 * $T = \left({S, \tau}\right)$ 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.