TITLE: Examples of extremally disconnected spaces
QUESTION [9 upvotes]: I am trying to understand the notion of extremally disconnected space
(in other words Stonean space),
i.e. a space in which any open set has an open closure.
Could you help me and give (reasonable) examples of such spaces except discrete space?
Thanks in advance.

REPLY [3 votes]: Two simple examples of extremally disconnected spaces are $\mathbb{N}$ (or any infinite set) with the cofinite topology $\mathcal{T}=\{A:|\mathbb{N}\setminus A|<\aleph_0\}\cup\{\varnothing\}$ and $\mathbb{R}$ with the right order topology $\mathcal{T}=\{(a,\infty):a\in\mathbb{R}\}\cup\{\varnothing,\mathbb{R}\}$.  In each, the closure of any nonempty open set is the whole space, hence open, and of course, every topological space satisfies $\overline\varnothing=\varnothing$, which is also open.
Note. Some texts such as Steen and Seebach's Counterexamples in Topology (SS) restrict the definition of extremal disconnectedness to Hausdorff spaces, which the above two are not. They are hyperconnected spaces, all of which are extremally disconnected (without the Hausdorff restriction).
The Hausdorff restriction limits the number of extremally disconnected spaces in SS to just these five:

Discrete Space
Stone–Čech Compactification of the Integers
Strong Ultrafilter Topology
Single Ultrafilter Topology
Sierpiński's Metric Space