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Stone's duality

In mathematics, Stone's duality, named in honor of Marshall H. Stone, is the duality between the category of Boolean algebras and the category of Stone spaces, i.e., totally disconnected compact Hausdorff topological spaces. In the category of Boolean algebras, the morphisms are Boolean homomorphisms. In the category of Stone spaces, the morphisms are continuous functions. Stone's duality generalises to infinite sets of propositions the use of 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 pointwise convergence of nets of such homomorphisms. Every Boolean algebra is isomorphic to the algebra of 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 homeomorphic to the space of 2-valued homomorphisms on 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 xS or x not ∈ S. (Perhaps this is one of the few occasions for such rapid-fire mulitple repetition of the two distinct words homomorphism and 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.