A Stone space, also called a Boolean space, is a topological space that is zero-dimensional, (http://planetmath.org/T0Space) and compact. Equivalently, a Stone space is a totally disconnected compact Hausdorff space.
Given a stone space , one may associate a Boolean algebra by taking the set of all of its clopen sets. The set theoretic operations of intersection, union, and complement makes a Boolean algebra. is known as the dual algebra of .
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 bijective correspondence between the following
- 1 Paul R. Halmos, Lectures on Boolean Algebras, D. Van Nostrand Company, Inc., 1963.
- 2 Peter T. Johnstone, Stone Spaces, Cambridge University Press, 1982.
|Date of creation||2013-03-22 13:24:23|
|Last modified on||2013-03-22 13:24:23|
|Last modified by||CWoo (3771)|