Editing Stone space

{{Short description|Type of topological space}}
{{More citations needed|date=December 2024}}
In [[topology]] and related areas of [[mathematics]], a '''Stone space''', also known as a '''profinite space<ref name=":0">{{nlab|id=Stone+space|title=Stone space}}</ref>''' or '''profinite set''', is a [[compact space|compact]] [[Hausdorff space|Hausdorff]] [[totally disconnected]] space.<ref name=":1">{{SpringerEOM
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}}</ref> Stone spaces are named after [[Marshall Harvey Stone]] who introduced and studied them in the 1930s in the course of his investigation of [[Boolean algebra (structure)|Boolean algebras]], which culminated in [[Stone's representation theorem for Boolean algebras|his representation theorem for Boolean algebras]].

== Equivalent conditions ==
The following conditions on the topological space <math>X</math> are equivalent:<ref name=":1" /><ref name=":0" />

* <math>X</math> is a Stone space;
* <math>X</math> is [[Homeomorphism|homeomorphic]] to the [[Inverse limit|projective limit]] (in the [[category of topological spaces]]) of an inverse system of finite [[discrete space]]s;
* <math>X</math> is compact and [[Connected space#Disconnected spaces|totally separated]];
* <math>X</math> is compact, [[Kolmogorov space|T<sub>0</sub>]], and [[Zero-dimensional space|zero-dimensional]] (in the sense of the [[small inductive dimension]]);
* <math>X</math> is [[Spectral space|coherent]] and Hausdorff.

== Examples ==
Important examples of Stone spaces include finite [[discrete space]]s, the [[Cantor set]] and the space <math>\Z_p</math> of [[P-adic integers|<math>p</math>-adic integers]], where <math>p</math> is any [[prime number]].  Generalizing these examples, any [[Product topology|product]] of arbitrarily many finite discrete spaces is a Stone space, and the topological space underlying any [[profinite group]] is a Stone space. The [[Stone–Čech compactification]] of the natural numbers with the discrete topology, or indeed of any discrete space, is a Stone space.

== Stone's representation theorem for Boolean algebras ==
{{main|Stone's representation theorem for Boolean algebras}}

To every [[Boolean algebra (structure)|Boolean algebra]] <math>B</math> we can associate a Stone space <math>S(B)</math> as follows: the elements of <math>S(B)</math> are the [[ultrafilter]]s on <math>B,</math> and the topology on <math>S(B),</math> called {{em|{{visible anchor|the Stone topology|Stone topology}}}}, is generated by the sets of the form <math>\{ F \in S(B) : b \in F \},</math> where <math>b \in B.</math>

[[Stone's representation theorem for Boolean algebras]] states that every Boolean algebra is isomorphic to the Boolean algebra of [[clopen set]]s of the Stone space <math>S(B)</math>; and furthermore, every Stone space <math>X</math> is homeomorphic to the Stone space belonging to the Boolean algebra of clopen sets of <math>X.</math> These assignments are [[functor]]ial, and we obtain a [[Dual category|category-theoretic duality]] between the category of Boolean algebras (with homomorphisms as morphisms) and the category of Stone spaces (with continuous maps as morphisms).

Stone's theorem gave rise to a number of similar dualities, now collectively known as [[Stone duality|Stone dualities]].

== Condensed mathematics ==
The category of Stone spaces with continuous maps is [[Equivalence of categories|equivalent]] to the [[pro-category]] of the [[category of finite sets]], which explains the term "profinite sets". The profinite sets are at the heart of the project of [[condensed mathematics]], which aims to replace topological spaces with "condensed sets", where a topological space ''X'' is replaced by the [[functor]] that takes a profinite set ''S'' to the set of continuous maps from ''S'' to ''X''.<ref>{{Cite web|last=Scholze|first=Peter|date=2020-12-05|title=Liquid tensor experiment|url=https://xenaproject.wordpress.com/2020/12/05/liquid-tensor-experiment/|website=Xena|language=en}}</ref>

== See also ==

* {{annotated link|Stone–Čech compactification#Construction using ultrafilters}}
* {{annotated link|Filters in topology}}
* {{annotated link|Type (model theory)}}

== References ==

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== Further reading ==

*{{cite book |author-link=Peter Johnstone (mathematician) |first=Peter |last=Johnstone |title=Stone Spaces |publisher=Cambridge University Press |date=1982 |series=Cambridge studies in advanced mathematics |volume=3 |isbn=0-521-33779-8 |url={{GBurl|CiWwoLNbpykC|pg=PR5}}}}

[[Category:Boolean algebra]]
[[Category:Categorical logic]]
[[Category:General topology]]