Definition:Stone Space

Definition

Let $S$ be a topological space.

Let $S$ be:

compact

Hausdorff

totally disconnected.

Then $S$ is a Stone space.

Boolean Lattice

Let $B$ be a Boolean lattice.

The Stone space of $B$ is the topological space:

$\map S B = \struct {U, \tau}$

where:

$(1): \quad U$ is the set of ultrafilters in $B$

$(2): \quad \tau$ is the topology generated by the basis consisting of all sets of the form:

$\exists b \in B: \set {x \in U: b \in x}$

Also see

• Results about Stone spaces can be found here.

Source of Name

This entry was named for Marshall Harvey Stone.

Sources

• 1987: S.B. Niefield and K.I. Rosenthal: Sheaves of Integral Domains on Stone Spaces (J. Pure Appl. Algebra Vol. 47: pp. 173 – 179)
  Zentralblatt MATH: 0631.13018