# Extremally Disconnected by Interior of Closed Sets

## Contents

## 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 $T = \struct {S, \tau}$ 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 $T = \struct {S, \tau}$ is **extremally disconnected** if and only if the interior of every closed set of $T$ is closed.

## Proof

Let $T = \struct {S, \tau}$ 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 $\paren {S \setminus V}^-$ is open by hypothesis.

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

- $\paren {S \setminus V}^- = S \setminus V^\circ$

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.

$\blacksquare$

## 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