# Extremally Disconnected Space is Totally Separated

## Theorem

Let $T = \struct {S, \tau}$ be an extremally disconnected topological space.

Then $T$ is totally separated.

## Proof

Let $T = \struct {S, \tau}$ 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, $U^- \cap V = \O$, so $y \notin U^-$.

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

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

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness: Disconnectedness