In %20%20%20Mathematics%20is%20commonly%20defined%20as%20the%20study%20of%20patterns%20of%20structure%2C%20change%2C%20and%20space%3B%20more%20informally%2C%20one%20might%20say%20it%20is%20the%20study%20of%20%5Cfigures%20and%20numbers%5C.%20Mathematical%20knowledge%20is%20constantly%20growing%2C%20through%20research%20and%20application%2C%20but%20mathematics%20itself%20is%20not%20usually%20considered%20a%20natural%20science.%20... mathematics, Stone's representation theorem for Boolean algebras, named in honor of %20%20Marshall%20Harvey%20Stone%20(April%208%2C%201903%26%238211%3BJanuary%209%2C%201989)%20was%20a%20American%20mathematician%20who%20made%20several%20important%20contributions%20in%20various%20areas%20of%20mathematical%20analysis%2C%20including%20in%20particular%20functional%20analysis.%20... Marshall H. Stone, is the %20%20In%20category%20theory%2C%20an%20abstract%20branch%20of%20mathematics%2C%20an%20equivalence%20of%20categories%20is%20a%20relation%20between%20two%20categories%20that%20establishes%20that%20these%20categories%20are%20%5Cessentially%20the%20same%5C.%20There%20are%20numerous%20examples%20of%20categorical%20equivalences%20from%20many%20areas%20of%20mathematics.%20... duality between the %20%20%20Category%20theory%20is%20a%20mathematical%20theory%20that%20deals%20in%20an%20abstract%20way%20with%20mathematical%20structures%20and%20relationships%20between%20them.%20... category of %20%20In%20mathematics%20and%20computer%20science%2C%20Boolean%20algebras%2C%20or%20Boolean%20lattices%2C%20are%20algebraic%20structures%20which%20%5Ccapture%20the%20essence%5C%20of%20the%20logical%20operations%20AND%2C%20OR%20and%20NOT%20as%20well%20as%20the%20corresponding%20set%20theoretic%20operations%20intersection%2C%20union%20and%20complement.%20... Boolean algebras and the category of Stone spaces, i.e., %20%20In%20topology%20and%20related%20branches%20of%20mathematics%2C%20a%20topological%20space%20X%20is%20said%20to%20be%20disconnected%20if%20it%20is%20the%20union%20of%20two%20disjoint%20nonempty%20open%20sets.%20... totally disconnected %20%20%20In%20mathematics%2C%20a%20compact%20space%20is%20a%20space%20that%20resembles%20a%20closed%20and%20bounded%20subset%20of%20Euclidean%20space%20Rn%20in%20that%20it%20is%20%5Csmall%5C%20in%20a%20certain%20sense%20and%20%5Ccontains%20all%20its%20limit%20points%5C.%20The%20modern%20general%20definition%20calls%20a%20topological%20space%20compact%20if%20every%20open%20cover%20of%20it%20has... compact %20%20%20In%20topology%20and%20related%20branches%20of%20mathematics%2C%20a%20Hausdorff%20space%20is%20a%20topological%20space%20in%20which%20points%20can%20be%20separated%20by%20neighbourhoods.%20... Hausdorff %20%20%20Topological%20spaces%20are%20structures%20that%20allow%20one%20to%20formalize%20concepts%20such%20as%20convergence%2C%20connectedness%20and%20continuity.%20... topological spaces. It is a special case of %20%20%20In%20mathematics%2C%20especially%20in%20topology%20and%20order%20theory%2C%20there%20is%20an%20ample%20supply%20of%20categorical%20dualities%20between%20certain%20categories%20of%20topological%20spaces%20and%20categories%20of%20partially%20ordered%20sets.%20... Stone duality, a general framework for dualities between topological spaces and partially ordered sets. In the category of Boolean algebras, the morphisms are %20%20%20A%20Boolean%20homomorphism%20is%20a%20homomorphism%20from%20one%20Boolean%20algebra%20into%20another.%20%20%20... Boolean homomorphisms. In the category of Stone spaces, the morphisms are %20%20In%20topology%2C%20a%20continuous%20function%20is%20generally%20defined%20as%20one%20for%20which%20preimages%20of%20open%20sets%20are%20open.%20... continuous functions. Stone's duality generalises to infinite sets of propositions the use of %20%20%20Truth%20tables%20are%20a%20type%20of%20mathematical%20table%20used%20in%20logic%20to%20determine%20whether%20an%20expression%20is%20true%20or%20whether%20an%20argument%20is%20valid.%20... truth tables to characterise elements of finite Boolean algebras. It employs systematically the two-element Boolean algebra {0,1} or {F,T} of truth-values, as the target of homomorphisms; this algebra may be written simply as 2.

In detail, the Stone space of a Boolean algebra A is the set of all 2-valued homomorphisms on A, with the topology of %20%20%20Suppose%20%7B%20fn%20%7D%20is%20a%20sequence%20of%20functions%20sharing%20the%20same%20domain%20in%20common%20(for%20the%20moment%2C%20we%20defer%20making%20precise%20the%20nature%20of%20the%20values%20of%20these%20functions%2C%20but%20the%20reader%20may%20take%20them%20to%20be%20real%20numbers%20if%20that%20makes%20anyone%20feel%20good).%20... pointwise convergence of %20%20%20In%20mathematics%20the%20term%20net%20has%20at%20least%20two%20meanings.%20... nets of such homomorphisms. (An alternative and equivalent way to construct the Stone space of A is as the set of all %20%20%20In%20mathematics%2C%20especially%20in%20order%20theory%2C%20an%20ultrafilter%20is%20a%20subset%20of%20a%20partially%20ordered%20set%20(a%20poset)%20which%20is%20maximal%20among%20all%20proper%20filters.%20... ultrafilters in A, with the sets {U : U is an ultrafilter containing a} for a in A as %20%20%20In%20mathematics%2C%20a%20base%20(or%20basis)%20B%20for%20a%20topological%20space%20X%20with%20topology%20T%20is%20a%20collection%20of%20open%20sets%20in%20T%20such%20that%20every%20open%20set%20in%20T%20can%20be%20written%20as%20a%20union%20of%20elements%20of%20B.%20... base of the topology. In the sequel we will use the homomorphism approach.)

Every Boolean algebra is isomorphic to the algebra of %20%20In%20topology%2C%20a%20clopen%20set%20(or%20closed-open%20set)%20in%20a%20topological%20space%20is%20a%20set%20which%20is%20both%20open%20and%20closed.%20%20... clopen (i.e., simultaneously closed and open) subsets of its Stone space. The isomorphism maps any element a of A to the set of homomorphisms that map a to 1.

Every totally disconnected compact Hausdorff space is %20%20%20%20This%20word%20should%20not%20be%20confused%20with%20homomorphism.%20%20%20... homeomorphic to the Stone space of the Boolean algebra of all of its clopen subsets. The homeomorphism maps each point x to the 2-valued homomorphism φ given by φ(S) = 1 or 0 according as x ∈ S or x not ∈ S. (Perhaps this is one of the few occasions for such rapid-fire multiple repetition of the two distinct words %20%20%20%20This%20word%20should%20not%20be%20confused%20with%20homeomorphism.%20%20%20... homomorphism and %20%20%20%20This%20word%20should%20not%20be%20confused%20with%20homomorphism.%20%20%20... homeomorphism in one breath. Let us therefore warn the reader not to confuse them with each other.)

Homomorphisms from a Boolean algebra A to a Boolean algebra B correspond in a natural way to continuous functions from the Stone space of B into the Stone space of A. In other words, this duality is a contravariant functor.

The Stone representation theorem cannot be proven within the %20%20The%20Zermelo-Fraenkel%20axioms%20of%20set%20theory%20(ZF)%20are%20the%20standard%20axioms%20of%20axiomatic%20set%20theory%20on%20which%2C%20together%20with%20the%20axiom%20of%20choice%2C%20all%20of%20ordinary%20mathematics%20is%20based%20in%20modern%20formulations.%20... Zermelo-Fraenkel axioms. It is equivalent to the %20%20%20In%20mathematics%2C%20a%20number%20of%20prime%20ideal%20theorems%20for%20guaranteeing%20the%20existence%20of%20certain%20subsets%20of%20an%20abstract%20algebra%20can%20be%20stated.%20... Boolean prime ideal theorem which states that every Boolean algebra has a prime ideal. Both can be proven using the %20%20%20In%20%20%20mathematics%2C%20the%20axiom%20of%20choice%20is%20an%20%20%20axiom%20of%20%20%20set%20theory.%20... axiom of choice. But the Stone representation theorem is strictly weaker than the axiom of choice.

This theorem was proved by %20%20Marshall%20Harvey%20Stone%20(April%208%2C%201903%26%238211%3BJanuary%209%2C%201989)%20was%20a%20American%20mathematician%20who%20made%20several%20important%20contributions%20in%20various%20areas%20of%20mathematical%20analysis%2C%20including%20in%20particular%20functional%20analysis.%20... Marshall H. Stone in %20%20%20%201934%20was%20a%20%20%20common%20year%20starting%20on%20Monday%20(link%20will%20take%20you%20to%20calendar).%20%20%20... 1934. His interest in these questions arose from his study of the %20%20%20%20%20%20The%20noun%20spectrum%20(plural%3A%20spectra)%20has%20a%20variety%20of%20meanings.%20... spectral theory of %20%20In%20mathematics%2C%20a%20linear%20transformation%20(also%20called%20linear%20operator%20or%20linear%20map)%20is%20a%20function%20between%20two%20vector%20spaces%20that%20respects%20the%20arithmetical%20operations%20addition%20and%20scalar%20multiplication%20defined%20on%20vector%20spaces%2C%20or%2C%20in%20other%20words%2C%20it%20%5Cpreserves%20linear%20combinations%5C.%20%20%20Definition%20and%20first%20consequences%20Formally%2C%20if%20V%20and%20W%20are... operators on a %20%20%20In%20mathematics%2C%20a%20Hilbert%20space%20is%20an%20inner%20product%20space%20that%20is%20complete%20with%20respect%20to%20the%20norm%20defined%20by%20the%20inner%20product.%20... Hilbert space. Stone's theorem has since been the model for many other similar representation theorems.

See also

• %20%20%20George%20Boole%20%20Boolean%20algebra%20%20Boolean%20function%20%20Boolean%20homomorphism%20%20Boolean%20prime%20ideal%20theorem%20%20Boolean%20satisfiability%20problem%20%20canonical%20form%20(Boolean%20algebra)%20%20compactness%20theorem%20%20connective%20--%20see%20logical%20operator%20de%20Morgan%5Cs%20laws%20%20Augustus%20De%20Morgan%20%20duality%20(order%20theory)%20%20formal%20system%20%20Heyting%20algebra%20%20interior%20algebra%20%20Karnaugh%20map%20%20Lindenbaum%20algebra%20%20logic%20gate%20%20logical%20operator%20%20monadic... list of Boolean algebra topics
• %20%20%20In%20mathematics%20a%20field%20of%20sets%20is%20a%20pair%20%20where%20%20is%20a%20set%20and%20%20is%20an%20algebra%20over%20%20i.... Field of sets
• %20%20%20In%20mathematics%2C%20especially%20in%20topology%20and%20order%20theory%2C%20there%20is%20an%20ample%20supply%20of%20categorical%20dualities%20between%20certain%20categories%20of%20topological%20spaces%20and%20categories%20of%20partially%20ordered%20sets.%20... Stone duality