Extremally Disconnected by Disjoint Open 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 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