# Extremally Disconnected by Disjoint Open Sets

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## 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 $T = \left({S, \tau}\right)$ is **extremally disconnected** if and only if the closure of every open set of $T$ is open.

### Definition 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

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 4$: Disconnectedness