Editing Stone space (section)

== 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]].