Stone space

The following conditions on the topological space X are equivalent:[2][1]

• X is a Stone space;
• X is homeomorphic to the projective limit (in the category of topological spaces) of an inverse system of finite discrete spaces;
• X is compact and totally separated;
• X is compact, T₀ , and zero-dimensional (in the sense of the small inductive dimension);
• X is coherent and Hausdorff.

Important examples of Stone spaces include finite discrete spaces, the Cantor set and the space Z_p of p-adic integers, where p is any prime number. Generalizing these examples, any product of finite discrete spaces is a Stone space, and the topological space underlying any profinite group is a Stone space. The Stone–Čech compactification of the natural numbers with the discrete topology, or indeed of any discrete space, is a Stone space.

To every Boolean algebra B we can associate a Stone space S(B) as follows: the elements of S(B) are the ultrafilters on B, and the topology on S(B), called the Stone topology, is generated by the sets of the form {F∈S(B) : b∈F}, where b is an element of B.

Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to the Boolean algebra of clopen sets of the Stone space S(B); and furthermore, every Stone space X is homeomorphic to the Stone space belonging to the Boolean algebra of clopen sets of X. These assignments are functorial, and we obtain a category-theoretic duality between the category of Boolean algebras (with homomorphisms as morphisms) and the category of Stone spaces (with continuous maps as morphisms).

Stone's theorem gave rise to a number of similar dualities, now collectively known as Stone dualities.

• Peter Johnstone, Stone Spaces, Cambridge University Press, 1982

• Stone–Čech compactification#Construction using ultrafilters