Extremally Disconnected by Interior of Closed Sets

Proof
Let $T = \left({S, \tau}\right)$ be a $T_2$ (Hausdorff) topological space such that the closure of every open set of $T$ is open.

Let $V \subseteq S$ be closed in $T$.

Then $S \setminus V$ is open by definition.

Then its closure $\left({S \setminus V}\right)^-$ is open by hypothesis.

By Complement of Interior equals Closure of Complement we have that:
 * $\left({S \setminus V}\right)^- = S \setminus V^\circ$

where $V^\circ$ is the interior of $V$.

As $S \setminus V^\circ$ is open in $T$, it follows that $V^-$ is closed.

So the interior of every closed set of $T$ is closed.

By a similar argument we see that if the interior of every closed set of $T$ is closed in $T$, then the closure of every open set of $T$ is open.

Hence the result.