Definition:T4 Space

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

$T = \left({X, \vartheta}\right)$ is a $T_4$ space iff:


 * $\forall A, B \in \complement \left({\vartheta}\right), A \cap B = \varnothing: \exists U, V \in \vartheta: A \subseteq U, B \subseteq V, U \cap V = \varnothing$

That is, for any two disjoint closed sets $A, B \subseteq X$ there exist disjoint open sets $U, V \in \vartheta$ containing $A$ and $B$ respectively.

That is:
 * $T = \left({X, \vartheta}\right)$ is $T_4$ when any two disjoint closed subsets of $X$ are separated by neighborhoods.

Equivalent Definitions
$T = \left({X, \vartheta}\right)$ is $T_4$ iff each open set $U$ contains a closed neighborhood of each closed set contained in $U$.

This is proved in Equivalent Definitions for $T_4$ Space.