Stone representation theorem

E924199

mathematical theorem representation theorem result in mathematical logic result in topology

The Stone representation theorem is a fundamental result in mathematical logic and topology that represents every Boolean algebra as an algebra of clopen sets in a totally disconnected compact Hausdorff (Stone) space.

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Surface form	Occurrences
Stone representation theorems	1

Statements (48)

Predicate	Object
instanceOf	mathematical theorem ⓘ representation theorem ⓘ result in mathematical logic ⓘ result in topology ⓘ
alsoKnownAs	Stone duality for Boolean algebras NERFINISHED ⓘ
appliesTo	finite Boolean algebras ⓘ infinite Boolean algebras ⓘ
associatedSpace	Stone space of ultrafilters of a Boolean algebra GENERATED ⓘ
categoryOnAlgebraSide	category of Boolean algebras with Boolean homomorphisms ⓘ
categoryOnSpaceSide	category of Stone spaces with continuous maps ⓘ
codomain	topological spaces ⓘ
concerns	Boolean algebras NERFINISHED ⓘ Boolean spaces ⓘ Stone spaces ⓘ
construction	associates to each Boolean algebra the space of its ultrafilters with the Stone topology ⓘ
domain	algebraic structures ⓘ
field	lattice theory ⓘ mathematical logic ⓘ topology ⓘ universal algebra ⓘ
generalizedBy	Priestley duality NERFINISHED ⓘ Stone duality for distributive lattices NERFINISHED ⓘ Stone-type dualities in topos theory ⓘ
guarantees	the Stone space of a Boolean algebra is Hausdorff ⓘ the Stone space of a Boolean algebra is compact ⓘ the Stone space of a Boolean algebra is totally disconnected ⓘ
implies	every Boolean algebra can be represented as a Boolean algebra of sets ⓘ
importance	establishes a duality between algebraic and topological structures ⓘ provides a concrete set-theoretic representation of abstract Boolean algebras ⓘ
inspired	representation theorems in lattice theory ⓘ
involves	Galois connection between ideals and open sets ⓘ
namedAfter	Marshall Harvey Stone NERFINISHED ⓘ
relates	Boolean algebra homomorphisms and continuous maps between Stone spaces ⓘ
specialCaseOf	Stone duality NERFINISHED ⓘ
statement	Every Boolean algebra is isomorphic to a field of sets. ⓘ Every Boolean algebra is isomorphic to an algebra of clopen subsets of a Stone space. ⓘ There is a dual equivalence between the category of Boolean algebras and the category of Stone spaces. ⓘ
typeOfDuality	contravariant equivalence of categories ⓘ
usedIn	measure theory on Boolean algebras ⓘ model theory ⓘ set-theoretic topology ⓘ theory of Boolean-valued models ⓘ
usesConcept	Boolean homomorphisms ⓘ clopen sets ⓘ compact Hausdorff space ⓘ totally disconnected space ⓘ ultrafilters ⓘ
yearProved	1936 ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Gelfand–Naimark theorem → isRelatedTo → Stone representation theorem ⓘ

Banach–Mazur theorem → relatedTo → Stone representation theorem ⓘ

this entity surface form: Stone representation theorems