Extremally disconnected space

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. The term "extremely disconnected" is sometimes used, but it is incorrect.)
An extremally disconnected space that is also compact and Hausdorff is sometimes called a Stonean space. This is different from a Stone space, which is usually a totally disconnected compact Hausdorff space. 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).
Equivalent characterizations.
A theorem due to says that the projective objects of the category of compact Hausdorff spaces are exactly the extremally disconnected compact Hausdorff spaces. A simplified proof of this fact is given by .
A compact Hausdorff space is extremally disconnected if and only if it is a retract of the Stone–Čech compactification of a discrete space.
Applications.
 proves the Riesz–Markov–Kakutani representation theorem by reducing it to the case of extremally disconnected spaces, in which case the representation theorem can be proved by elementary means.