Stone space

A Stone space, also called a Boolean space, is a topological spaceMathworldPlanetmath that is zero-dimensional, T0 ( and compactPlanetmathPlanetmath. Equivalently, a Stone space is a totally disconnected compact Hausdorff space.

Given a stone space X, one may associate a Boolean algebraMathworldPlanetmath X* by taking the set of all of its clopen sets. The set theoretic operationsMathworldPlanetmath of intersectionMathworldPlanetmathPlanetmath, union, and complementPlanetmathPlanetmath makes X* a Boolean algebra. X* is known as the dual algebra of X.

The significance of Stone spaces stems from Stone duality: a pervasive equivalence between the algebraic notions and theorems of Boolean algebras on one hand, and the topological notions and theorems of Stone spaces on the other. This equivalence comprises the content and consequences of M. H. Stone’s representation theorem.

There is a bijectiveMathworldPlanetmathPlanetmath correspondence between the following

  1. 1.

    The class consisting of all Boolean spaces

  2. 2.

    The class consisting of all Boolean algebras

  3. 3.

    The class consisting of all Boolean ringsMathworldPlanetmath

  4. 4.

    The class consisting of all prime spectra of von Neumann regular ringsMathworldPlanetmath

In fact, viewing each class as a categoryMathworldPlanetmath equipped with the appropriate class of morphismsMathworldPlanetmath, the categories are naturally equivalent between one another.

More to come: partial proofs, constructions, categorical equivalence between the first three items due to Stone’s Representation Theorem and references .


  • 1 Paul R. Halmos, Lectures on Boolean Algebras, D. Van Nostrand Company, Inc., 1963.
  • 2 Peter T. Johnstone, Stone Spaces, Cambridge University Press, 1982.
Title Stone space
Canonical name StoneSpace
Date of creation 2013-03-22 13:24:23
Last modified on 2013-03-22 13:24:23
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 19
Author CWoo (3771)
Entry type Definition
Classification msc 06B30
Classification msc 06E15
Synonym Boolean space
Related topic DualityInMathematics
Related topic DualOfStoneRepresentationTheorem
Defines dual algebra