Extremally disconnected space


In mathematics, an extremally disconnected space is a topological space in which the closure of every open set is open.
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 is equivalent to the property of being discrete.

Examples

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.