Equivalence of Definitions of T5 Space

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

The following two conditions defining a $T_5$ space are logically equivalent:

Definition by Open Sets
$T$ is a $T_5$ space iff:


 * $\forall A, B \subseteq X, A^- \cap B = A \cap B^- = \varnothing: \exists U, V \in \vartheta: A \subseteq U, B \subseteq V, U \cap V = \varnothing$

That is:
 * $\left({X, \vartheta}\right)$ is a $T_5$ space when for any two separated sets $A, B \subseteq X$ there exist disjoint open sets $U, V \in \vartheta$ containing $A$ and $B$ respectively.

Definition by Closed Neighborhoods
$\left({X, \vartheta}\right)$ is a $T_5$ space iff each subset $Y$ contains a closed neighborhood of each $A \subseteq Y^\circ$ where $A^- \subseteq Y$

where $Y^\circ$ denotes the interior of $Y$ and $Y^-$ denotes the closure of $Y$.