Equivalence of Definitions of T4 Space

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

The following two conditions defining a $T_4$ space are logically equivalent:

Definition by Open Sets
$T$ 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.

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