M. H. Stone’s representation theorem

Theorem 1.

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


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


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.


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,


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


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.

Title M. H. Stone’s representation theorem
Canonical name MHStonesRepresentationTheorem
Date of creation 2013-03-22 13:25:34
Last modified on 2013-03-22 13:25:34
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 19
Author rspuzio (6075)
Entry type Theorem
Classification msc 54D99
Classification msc 06E99
Classification msc 03G05
Synonym Stone representation theorem
Synonym Stone’s representation theorem
Related topic RepresentingABooleanLatticeByFieldOfSets
Related topic DualSpaceOfABooleanAlgebra