Order, lattices, ordered algebraic structures Boolean algebras Theorem

Stone duality

Builds on Boolean algebra Stone space Ultrafilter on a Boolean algebra Clopen set Equivalence of categories Stone's representation theorem for Boolean algebras

Statement

Let $\mathbf{Bool}$ be the category of Boolean algebras with Boolean algebra homomorphisms. Let $\mathbf{Stone}$ be the category of Stone spaces with continuous maps. Then $\mathbf{Bool}$ is equivalent to the opposite category of $\mathbf{Stone}$ , by the following functors.

A Boolean algebra $B$ goes to its space $S(B)$ of ultrafilters, with basic open sets $\set{U : a \in U}$ for $a \in B$ . A homomorphism $h \colon A \to B$ goes to the continuous map $S(B) \to S(A)$ , $U \mapsto h^{-1}(U)$ .
A Stone space $X$ goes to the Boolean algebra $\operatorname{Clop}(X)$ of its clopen subsets; a continuous map $f \colon X \to Y$ goes to the homomorphism $\operatorname{Clop}(Y) \to \operatorname{Clop}(X)$ , $C \mapsto f^{-1}(C)$ .
The maps $B \to \operatorname{Clop}(S(B))$ , $a \mapsto \set{U : a \in U}$ , and $X \to S(\operatorname{Clop}(X))$ , $x \mapsto \set{C : x \in C}$ , are natural isomorphisms.

Remarks

The first isomorphism in clause 3 is Stone's representation theorem. The second one holds for this reason. In a compact Hausdorff space with a basis of clopen sets, the members of an ultrafilter of clopen sets have exactly one common point. Under the duality, injective homomorphisms correspond to surjective continuous maps, and quotients of $B$ correspond to closed subspaces of $S(B)$ . For example, the free Boolean algebra on a set $I$ corresponds to the product space $\set{0, 1}^I$ , and the finite Boolean algebras correspond to the finite discrete spaces.

Going deeper

For a student

Stone duality lets one prove facts about Boolean algebras by topology and facts about Stone spaces by algebra. For instance, a Boolean algebra is complete exactly when its Stone space is extremally disconnected, and an atom of the algebra corresponds to an isolated point of the space. It is the model for many later dualities: Priestley duality for distributive lattices, Esakia duality for Heyting algebras, and the duality between spatial frames and sober spaces.

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Algebra of sets concept
Boolean algebra concept
Clopen set concept
Equivalence of categories concept
Stone space concept
Ultrafilter on a Boolean algebra concept
Stone's representation theorem for Boolean algebras statement
Stone duality statement

Tag	`B9T33D5MJK`
URI	`m2://B9T33D5MJK`
Kind	statement, theorem
Writing	W0 stub
MSC	06E15, 18F70
Status	This copy of the site has no build data. Run `m2 status B9T33D5MJK`.
About	Boolean algebra, Stone space
Uses	Boolean algebra, Stone space, Ultrafilter on a Boolean algebra, Clopen set, Equivalence of categories, Stone's representation theorem for Boolean algebras
Changed	2 November 2026

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