Characterization of T0 Space by Closed Sets

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

Then
 * $T$ is a $T_0$ space
 * $\forall x, y \in S: x \ne y \implies$
 * $\left({\exists F \subseteq S: S}\right.$ is closed $\left. {\land\, x \in F \land y \notin F}\right)$
 * $\lor$
 * $\left({\exists F \subseteq S: S}\right.$ is closed $\left. {\land\, x \notin F \land y \in F}\right)$