Extremally Disconnected by Interior of Closed Sets

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Theorem

The following definitions of the concept of Extremally Disconnected Space are equivalent:

Definition using Closures of Open Sets

A $T_2$ (Hausdorff) topological space $T = \left({S, \tau}\right)$ is extremally disconnected if and only if the closure of every open set of $T$ is open.

Definition using Interiors of Closed Sets

A $T_2$ (Hausdorff) topological space $T = \left({S, \tau}\right)$ is extremally disconnected if and only if the interior of every closed set of $T$ is closed.


Proof

Let $T = \left({S, \tau}\right)$ be a $T_2$ (Hausdorff) topological space such that the closure of every open set of $T$ is open.

Let $V \subseteq S$ be closed in $T$.

Then $S \setminus V$ is open by definition.

Then its closure $\left({S \setminus V}\right)^-$ is open by hypothesis.

By Complement of Interior equals Closure of Complement we have that:

$\left({S \setminus V}\right)^- = S \setminus V^\circ$

where $V^\circ$ is the interior of $V$.

As $S \setminus V^\circ$ is open in $T$, it follows that $V^-$ is closed.

So the interior of every closed set of $T$ is closed.

$\Box$


By a similar argument we see that if the interior of every closed set of $T$ is closed in $T$, then the closure of every open set of $T$ is open.

Hence the result.

$\blacksquare$


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