Characterization of T0 Space by Closures of Singletons

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

Then
 * $T$ is $T_0$ space
 * $\forall x, y \in S: x \ne y \implies x \notin \left\{{y}\right\}^- \lor y \notin \left\{{x}\right\}^-$

where $\left\{{y}\right\}^-$ denotes the closure of $\left\{{y}\right\}$