In topology and related areas of mathematics, a Stone space, also known as a profinite space[1] or profinite set, is a compact Hausdorff totally disconnected space.
[2] Stone spaces are named after Marshall Harvey Stone who introduced and studied them in the 1930s in the course of his investigation of Boolean algebras, which culminated in his representation theorem for Boolean algebras.
are equivalent:[2][1] Important examples of Stone spaces include finite discrete spaces, the Cantor set and the space
The Stone–Čech compactification of the natural numbers with the discrete topology, or indeed of any discrete space, is a Stone space.
called the Stone topology, is generated by the sets of the form
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
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.
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.