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.