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In topology and related areas of mathematics, a Stone space, also known as a profinite space,[1] is a compact totally disconnected Hausdorff 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 representing Boolean algebras.

Contents

Equivalent conditionsEdit

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

ExamplesEdit

Important examples of Stone spaces include the Cantor set and the space Zp of p-adic integers, where p is any prime number. Generalizing the first example, any product of finite discrete spaces is a Stone space; generalizing the second example, the topological space underlying any profinite group is a Stone space. The Stone–Čech compactification of the natural numbers, or of any discrete space, is a Stone space.

Stone's representation theorem for Boolean algebrasEdit

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) is generated by the sets of the form {FS(B) : bF}, 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 functiorial, 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).

Further readingEdit

ReferencesEdit

  1. ^ a b Stone space in nLab
  2. ^ a b Hazewinkel, Michiel, ed. (2001) [1994], "Stone space", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4