Extremally disconnected space

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In mathematics, a topological space is termed extremally disconnected or extremely disconnected if the closure of every open set in it is open. (The word "extremally" does not appear in most dictionaries^[1] and originated as a misspelling of "extremely"^{[citation needed]}, but is now the most common spelling for "extremally disconnected".)

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. Just as there is a duality between Stone spaces and Boolean algebras, there is a duality between Stonean spaces and the category of 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).

[edit] 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.

[edit] References

• A. V. Arkhangelskii (2001), “Extremally-disconnected space”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Johnstone, Peter T (1982). Stone spaces. CUP. ISBN 0521238935.