Equivalence of Definitions of Extremally Disconnected Space

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Theorem

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

Definition 1: 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 2: 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.

Definition 3: using Disjoint Open Sets

A $T_2$ (Hausdorff) topological space $T = \left({S, \tau}\right)$ is extremally disconnected if and only if the closures of every pair of open sets which are disjoint are also disjoint.


Proof

$(1)$ iff $(2)$

Extremally Disconnected by Interior of Closed Sets

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.

$\Box$


$(1)$ iff $(3)$

Extremally Disconnected by Disjoint Open Sets



$\blacksquare$