Discrete Space is Extremally Disconnected

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Theorem

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

$\blacksquare$


Sources