Extremally Disconnected Space is Totally Separated

Theorem
Let $T = \left({X, \vartheta}\right)$ be an extremally disconnected topological space.

Then $T$ is totally separated.

Proof
Let $T = \left({X, \vartheta}\right)$ be an extremally disconnected topological space.

Then by definition $T$ is a $T_2$ (Hausdorff) space such that the closure of every open set of $T$ is open.

Let $x, y \in X: x \ne y$.

As $T$ is a $T_2$ (Hausdorff) space, there exist disjoint open sets $U, V \in \vartheta$ containing $x$ and $y$ respectively.

From Closure is Closed, $\operatorname{cl}(U)$ is closed.

But $U$ is open, so from the definition of extremally disconnected space, then $\operatorname{cl}(U)$ is also open.

So $\operatorname{cl}(U)$ is clopen.

Even more: from Disjoint Open Sets remain Disjoint with one Closure $\operatorname{cl}(U)\cap V=\varnothing$, so $y\notin \operatorname{cl}(U)$.

This implies that $\left\{{\operatorname{cl}(U) \mid S\setminus \operatorname{cl}(U)}\right\}$ is a partition where $x\in\operatorname{cl}(U)$ and $y\in S\setminus \operatorname{cl}(U)$.