Extremally disconnected space

In mathematics, a topological space is termed extremally disconnected if the closure of every open set in it is open. (The term "extremally disconnected" is correct, even though the word "extremally" does not appear in most dictionaries.^[1] 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. (Note that this is different from a Stone space, which is usually a totally disconnected compact Hausdorff space.) A theorem due to Andrew Gleason says that the projective objects of the category of compact Hausdorff spaces are exactly the extremally disconnected compact Hausdorff spaces. 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

• Every discrete space is extremally disconnected.
• The Stone–Čech compactification of a discrete space is extremally disconnected.
• The spectrum of an abelian von Neumann algebra is extremally disconnected.
• Any set with the cofinite topology is extremally disconnected, but if the set is infinite this space is connected.

References

• A. V. Arkhangelskii (2001), "Extremally-disconnected space", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
• Johnstone, Peter T (1982). Stone spaces. CUP. ISBN 0-521-23893-5.