Discrete Space is Extremally Disconnected

Theorem
Let $T = \left({S, \vartheta}\right)$ be a topological space where $\vartheta$ is the discrete topology on $S$.

Then $T$ is extremally disconnected.

Proof
First we note that as Discrete Space Satisfies All Separation Properties, $T$ is a $T_2$ (Hausdorff) space.

Then from Interior Equals Closure of Subset of Discrete Space, it follows directly that the closure of every open set of $T$ is open.

Hence by definition $T$ is extremally disconnected.