Important examples of Stone spaces include finite discrete spaces, the Cantor set and the space of -adic integers, where is any prime number. Generalizing these examples, any product of arbitrarily many 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.
Stone's representation theorem for Boolean algebras
To every Boolean algebra we can associate a Stone space as follows: the elements of are the ultrafilters on and the topology on called the Stone topology, is generated by the sets of the form where
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 ; and furthermore, every Stone space is homeomorphic to the Stone space belonging to the Boolean algebra of clopen sets of 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.
Condensed mathematics
The category of Stone spaces with continuous maps is equivalent to the pro-category of the category of finite sets, which explains the term "profinite sets". The profinite sets are at the heart of the project of condensed mathematics, which aims to replace topological spaces with "condensed sets", where a topological space X is replaced by the functor that takes a profinite set S to the set of continuous maps from S to X.[3]
See also
Stone–Čech compactification#Construction using ultrafilters – a universal map from a topological space X to a compact Hausdorff space βX, such that any map from X to a compact Hausdorff space factors through βX uniquely; if X is Tychonoff, then X is a dense subspace of βXPages displaying wikidata descriptions as a fallback
Filters in topology – Use of filters to describe and characterize all basic topological notions and results.