Extremally Disconnected Space is Totally Separated

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Theorem

Let $T = \left({S, \tau}\right)$ be an extremally disconnected topological space.

Then $T$ is totally separated.


Proof

Let $T = \left({S, \tau}\right)$ 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 $x, y \in S: x \ne y$.

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

Let $U^-$ denote the closure of $U$.

From Topological Closure is Closed, $U^-$ is closed.

But $U$ is open, so from the definition of extremally disconnected space, then $U^-$ is also open.

So $U^-$ is clopen.


From Disjoint Open Sets remain Disjoint with one Closure, $U^- \cap V = \varnothing$, so $y \notin U^-$.

Thus $\left\{{U^- \mid S \setminus U^-}\right\}$ is a partition where $x \in U^-$ and $y \in X \setminus U^-$.

Thus, by definition, $T$ is totally separated.

$\blacksquare$


Sources