Extremally Disconnected by Disjoint Open Sets

Definition $1$ Definition $2$
Follows directly from Complement of Interior equals Closure of Complement.

Definition $1$ implies Definition $3$
Let $A, B \subseteq S$ be disjoint open sets.

Then:

As $A, B$ are arbitrary:


 * the closures of every pair of open sets which are disjoint are also disjoint.

Definition $3$ implies Definition $1$
Let $A \subseteq S$ be an open set.

By Topological Closure is Closed, $A^-$ is closed.

Hence $\relcomp S {A^-}$ is open.

We have:

By Set is Closed iff Equals Topological Closure, $\relcomp S {A^-}$ is closed.

Thus $\relcomp S {\relcomp S {A^-} } = A^-$ is open.

As $A$ is arbitrary:


 * the closure of every open set is open.