Stone space explained
In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact totally disconnected Hausdorff space. 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.
Equivalent conditions
The following conditions on the topological space
are equivalent:
is a Stone space;
is
homeomorphic to the
projective limit (in the
category of topological spaces) of an inverse system of finite
discrete spaces;
is compact and totally separated;
is compact,
T0, and
zero-dimensional (in the sense of the
small inductive dimension);
is
coherent and Hausdorff.
Examples
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
See main article: Stone's representation theorem for Boolean algebras.
we can associate a Stone space
as follows: the elements of
are the
ultrafilters on
and the topology on
called, 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.[1]
Further reading
- Book: Johnstone, Peter . Peter Johnstone (mathematician) . [{{GBurl|CiWwoLNbpykC|pg=PR5}} Stone Spaces ]. Cambridge University Press . 1982 . Cambridge studies in advanced mathematics . 3 . 0-521-33779-8 .
Notes and References
- Web site: Scholze. Peter. 2020-12-05. Liquid tensor experiment. Xena. en.
