View source for Extremally disconnected space

{{Short description|Topological space in which the closure of every open set is open}}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,<ref>{{Cite OED|extremally}}</ref> and is sometimes mistaken by spellcheckers for the [[homophone]] ''extremely&nbsp;disconnected''.)

An extremally disconnected space that is also [[compact space|compact]] and [[Hausdorff space|Hausdorff]] is sometimes called a '''Stonean space'''. This is not the same as a [[Stone space]], which is a [[totally disconnected]] compact Hausdorff space.  Every Stonean space is a Stone space, but not vice versa.  In the duality between Stone spaces and [[Boolean algebra (structure)|Boolean algebra]]s, the Stonean spaces correspond to the [[complete Boolean algebra]]s. 

An extremally disconnected [[first-countable space|first-countable]] [[collectionwise Hausdorff space]] must be [[discrete space|discrete]]. In particular, for [[metric 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 and non-examples==
* Every [[discrete space]] is extremally disconnected. Every [[indiscrete space]] is both extremally disconnected and connected.
* The [[Stone–Čech compactification]] of a discrete space is extremally disconnected.
* The [[spectrum (functional analysis)|spectrum]] of an [[abelian von Neumann algebra]] is extremally disconnected.
* Any commutative [[AW*-algebra]] is isomorphic to <math>C(X)</math>, for some space <math>X</math> which is extremally disconnected, compact and Hausdorff.
* Any infinite space with the [[cofinite topology]] is both extremally disconnected and [[connected space|connected]]. More generally, every [[hyperconnected space]] is extremally disconnected.
* The space on three points with [[base (topology)|base]] <math>\{\{x,y\},\{x,y,z\}\}</math> provides a [[finite set|finite]] example of a space that is both extremally disconnected and connected. Another example is given by the [[Sierpinski space]], since it is finite, connected, and hyperconnected.

The following spaces are not extremally disconnected:
* The [[Cantor set]] is not extremally disconnected. However, it is totally disconnected.

==Equivalent characterizations==
A theorem due to {{harvtxt|Gleason|1958}} says that the [[projective object]]s of the [[category (mathematics)|category]] of compact Hausdorff spaces are exactly the extremally disconnected compact Hausdorff spaces. A simplified proof of this fact is given by {{harvtxt|Rainwater|1959}}.

A compact Hausdorff space is extremally disconnected if and only if it is a [[retract (topology)|retract]] of the Stone–Čech compactification of a discrete space.<ref>{{harvtxt|Semadeni|1971|loc=Thm. 24.7.1}}</ref>

==Applications==

{{harvtxt|Hartig|1983}} 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.

== See also ==

* [[Totally disconnected space]]

== References ==
{{reflist}}
* {{springer|id=E/e037240|title=Extremally-disconnected space|author=[[Alexander Arhangelskii|A. V. Arkhangelskii]]}}
* {{Citation|last=Gleason|first=Andrew M.|author-link=Andrew Gleason|title=Projective topological spaces|journal=Illinois Journal of Mathematics|volume=2|issue=4A|year=1958|pages=482–489|mr=0121775|doi=10.1215/ijm/1255454110|doi-access=free}}
* {{Citation|author=Hartig|year=1983|first=Donald G.|title=The Riesz representation theorem revisited|journal=[[American Mathematical Monthly]]|volume=90|issue=4|pages=277&ndash;280|doi=10.2307/2975760|jstor=2975760 }}
* {{cite book| last = Johnstone
|authorlink = Peter Tennant Johnstone
  | first = Peter T.
  | title = Stone spaces
  | publisher = Cambridge University Press
  | date = 1982
  | isbn =0-521-23893-5}}
* {{Citation|doi=10.2307/2033466|jstor=2033466|title=A Note on Projective Resolutions|journal=Proceedings of the American Mathematical Society|volume=10|issue=5|pages=734–735|year=1959|last1=Rainwater|first1=John|authorlink = John Rainwater|doi-access=free}}
* {{Citation|last=Semadeni|first=Zbigniew|title=Banach spaces of continuous functions. Vol. I|publisher=PWN---Polish Scientific Publishers, Warsaw|year=1971|mr=0296671}}
[[Category:Properties of topological spaces]]