dual space of a Boolean algebra

Let B be a Boolean algebraMathworldPlanetmath, and B* the set of all maximal idealsMathworldPlanetmathPlanetmath of B. In this entry, we will equip B* with a topologyMathworldPlanetmath so it is a Boolean space.

Definition. For any aB, define M(a):={MB*aM}, and :={M(a)aB}.

It is know that in a Boolean algebra, maximal ideals and prime idealsMathworldPlanetmathPlanetmath coincide. From this entry (http://planetmath.org/RepresentingABooleanLatticeByFieldOfSets), we have the three following properties concerning M(a):


Furthermore, if M(a)=M(b), then a=b.

From these properties, we see that M(0)= and M(1)=B*. As a result, we see that

Proposition 1.

B* is a topological space, whose topology T is generated by the basis B.


and B* are both open, as they are M(0) and M(1) respectively. Also, the intersectionMathworldPlanetmathPlanetmath of open sets M(a) and M(b) is again open, since it is M(ab). ∎

We may in fact treat as a subbasis for 𝒯, since finite intersections of elements of remain in .

Proposition 2.

Each member of B is closed, hence T is generated by a basis of clopen sets. In other words, B* is zero-dimensional.


Each M(a) is open, by definition, and closed, since it is the complementPlanetmathPlanetmath of the open set M(a). ∎

Proposition 3.

B* is HausdorffPlanetmathPlanetmath.


If M,NB* such that MN, then there is some aB such that aM and aN. This means that NM(a) and MM(a), which means that MB*-M(a)=M(a). Since M(a) and M(a) are open and disjoint, with NM(a) and MM(a), we see that B* is Hausdorff. ∎

Now, based on a topological fact, every zero-dimensional Hausdorff space is totally disconnected. Hence B* is totally disconnected.

Proposition 4.

B* is compactPlanetmathPlanetmath.


Suppose {UiiI} is a collectionMathworldPlanetmath of open sets whose union is B*. Since each Ui is a union of elements of , we might as well assume that B* is covered by elements of . In other words, we may assume that each Ui is some M(ai).

Let J be the ideal generated by the set {aiiI}. If JB, then J can be extended to a maximal ideal M. Since each aiM, we see that MM(ai), so that M{M(ai)iI}=B*, which is a contradictionMathworldPlanetmathPlanetmath. Therefore, J=B. In particular, 1J, which means that 1 can be expressed as the join of a finite number of the ai’s:


where K is a finite subset of J. As a result, we have


So B* has a finite subcover, and hence is compact. ∎

Collecting the last three results, we see that B* is a Boolean space.

Remark. It can be shown that B is isomorphic to the Boolean algebra of clopen sets in B*. This is the famous Stone representation theorem.

Title dual spacePlanetmathPlanetmath of a Boolean algebra
Canonical name DualSpaceOfABooleanAlgebra
Date of creation 2013-03-22 19:08:35
Last modified on 2013-03-22 19:08:35
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 06E05
Classification msc 03G05
Classification msc 06B20
Classification msc 03G10
Classification msc 06E20
Related topic StoneRepresentationTheorem
Related topic MHStonesRepresentationTheorem