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单词 MHStonesRepresentationTheorem
释义

M. H. Stone’s representation theorem


Theorem 1.

Given a Boolean algebraMathworldPlanetmath B there exists a totally disconnected compactPlanetmathPlanetmathHausdorff space X such that B is isomorphic to the Boolean algebraof clopen subsets of X.

Proof.

Let X=B*, the dual spacePlanetmathPlanetmath (http://planetmath.org/DualSpaceOfABooleanAlgebra) of B, which is composed of all maximal idealsMathworldPlanetmathPlanetmath of B. According to this entry (http://planetmath.org/DualSpaceOfABooleanAlgebra), X is a Boolean space (totally disconnected compact Hausdorff) whose topologyMathworldPlanetmath is generated by the basis

:={M(a)aB},

where M(a)={MB*aM}.

Next, we show a general fact about the dual space B*:

Lemma 2.

is the set of all clopen sets in X.

Proof.

Clearly, every element of is clopen, by definition. Conversely, suppose U is clopen. Then U={M(ai)iI} for some index setMathworldPlanetmathPlanetmath I, since U is open. But U is closed, so B*-U={M(aj)jJ} for some index set J. Hence B*={M(ak)kIJ}. Since B* is compact, there is a finite subset K of IJ such that B*={M(ak)kK}. Let V={M(ai)iKI}. Then VU. But B*-VB*-U also. So U=V. Let y={aiiKI}, which exists because KI is finite. As a result,

U=V={M(ai)iKI}=M({aiiKI})=M(y).

Finally, based on the result of this entry (http://planetmath.org/RepresentingABooleanLatticeByFieldOfSets), B is isomorphic to the field of sets

F:={F(a)aB},

where F(a)={PP prime in B, and aP}. Realizing that prime idealsMathworldPlanetmathPlanetmath and maximal ideals coincide in any Boolean algebra, the set F is precisely .∎

Remark. There is also a dual version of the Stone representation theorem, which says that every Boolean space is homeomorphicMathworldPlanetmath to the dual space of some Boolean algebra.

TitleM. H. Stone’s representation theorem
Canonical nameMHStonesRepresentationTheorem
Date of creation2013-03-22 13:25:34
Last modified on2013-03-22 13:25:34
Ownerrspuzio (6075)
Last modified byrspuzio (6075)
Numerical id19
Authorrspuzio (6075)
Entry typeTheorem
Classificationmsc 54D99
Classificationmsc 06E99
Classificationmsc 03G05
SynonymStone representation theorem
SynonymStone’s representation theorem
Related topicRepresentingABooleanLatticeByFieldOfSets
Related topicDualSpaceOfABooleanAlgebra
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