- Extremally disconnected space
In
mathematics , atopological space is termed extremally disconnected or extremely disconnected if the closure of every open set in it is open. (The term "extremally disconnected" is usual, even though the word "extremally" does not appear in most dictionaries. [ [http://dictionary.oed.com/cgi/entry/50081102?nearest_to=extremally "extremally" in the O.E.D.] ] )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 atotally disconnected compact Hausdorff space.) A theorem due toAndrew Gleason says that theprojective object s 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 ofcomplete Boolean algebra s.An extremally disconnected first countable
collectionwise Hausdorff space must be discrete. In particular, formetric space s, 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.
* TheStone–Čech compactification of a discrete space is extremally disconnected.
* The spectrum of anabelian von Neumann algebra is extremally disconnected.References
*springer|id=E/e037240|title=Extremally-disconnected space|author=A. V. Arkhangelskii
*cite book| last = Johnstone
first = Peter T
title = Stone spaces
publisher = CUP
date = 1982
isbn =0521238935
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