Extremally disconnected space
topological space in which the closure of every open set is open
In mathematics, an extremally disconnected space is a topological space in which the closure of every open set is open. (The term "extremally disconnected" is correct, even though the word "extremally" does not appear in most dictionaries, and is sometimes mistaken by spellcheckers for the homophone extremely disconnected.)
An extremally disconnected space that is also compact and Hausdorff is sometimes called a Stonean space.
This is not the same as a Stone space, which is a totally disconnected compact Hausdorff space. Every Stonean space is a Stone space, but not vice versa. In the duality between Stone spaces and Boolean algebras, the Stonean spaces correspond to the complete Boolean algebras.
An extremally disconnected first-countable collectionwise Hausdorff space must be discrete. In particular, for metric spaces, the property of being extremally disconnected (the closure of every open set is open) is equivalent to the property of being discrete (every set is open).
Examples and non-examples
Every discrete space is extremally disconnected. Every indiscrete space is both extremally disconnected and connected.
The Stone–Čech compactification of a discrete space is extremally disconnected.
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