Stone's Representation Theorem For Boolean Algebras - Representation Theorem

Representation Theorem

A simple version of Stone's representation theorem states that any Boolean algebra B is isomorphic to the algebra of clopen subsets of its Stone space S(B). The isomorphism sends an element b∈B to the set of all ultrafilters that contain b. This is a clopen set because of the choice of topology on S(B) and because B is a Boolean algebra.

Restating the theorem using the language of category theory; the theorem states that there is a duality between the category of Boolean algebras and the category of Stone spaces. This duality means that in addition to the isomorphisms between Boolean algebras and their Stone spaces, each homomorphism from a Boolean algebra A to a Boolean algebra B corresponds in a natural way to a continuous function from S(B) to S(A). In other words, there is a contravariant functor that gives an equivalence between the categories. This was an early example of a nontrivial duality of categories.

The theorem is a special case of Stone duality, a more general framework for dualities between topological spaces and partially ordered sets.

The proof requires either the axiom of choice or a weakened form of it. Specifically, the theorem is equivalent to the Boolean prime ideal theorem, a weakened choice principle which states that every Boolean algebra has a prime ideal.