Stone representation theorem
E924199
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Stone representation theorem canonical | 1 |
| Stone representation theorems | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11411573 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Stone representation theorem Context triple: [Gelfand–Naimark theorem, isRelatedTo, Stone representation theorem]
-
A.
Khinchin's representation theorem
Khinchin's representation theorem is a result in probability theory that characterizes stationary stochastic processes by representing them in terms of simpler, more fundamental random components.
-
B.
Birkhoff’s representation theorem for finite distributive lattices
Birkhoff’s representation theorem for finite distributive lattices is a fundamental result in lattice theory that characterizes every finite distributive lattice as isomorphic to the lattice of lower (order) ideals of a finite poset.
-
C.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
-
D.
Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
-
E.
Birkhoff–von Neumann theorem
The Birkhoff–von Neumann theorem is a fundamental result in matrix theory and combinatorics stating that every doubly stochastic matrix can be expressed as a convex combination of permutation matrices.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Stone representation theorem Target entity description: 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.
-
A.
Khinchin's representation theorem
Khinchin's representation theorem is a result in probability theory that characterizes stationary stochastic processes by representing them in terms of simpler, more fundamental random components.
-
B.
Birkhoff’s representation theorem for finite distributive lattices
Birkhoff’s representation theorem for finite distributive lattices is a fundamental result in lattice theory that characterizes every finite distributive lattice as isomorphic to the lattice of lower (order) ideals of a finite poset.
-
C.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
-
D.
Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
-
E.
Birkhoff–von Neumann theorem
The Birkhoff–von Neumann theorem is a fundamental result in matrix theory and combinatorics stating that every doubly stochastic matrix can be expressed as a convex combination of permutation matrices.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Stone representation theorem Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Stone representation theorems