nLab
Stonean locale

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

A Stonean locale is an extremally disconnected Stone locale. The category of Stonean locales has open maps of locales as morphisms.

Thus, the category of Stonean locales is a (nonfull) subcategory of the category of Stone locales.

Properties

In presence of the axiom of choice, every Stonean locale is spatial and the category of Stonean locales is equivalent to the cateogry of Stonean spaces.

As a variant of Stone duality, the category of Stonean locales is contravariantly equivalent to the category of complete Boolean algebras and continuous homomorphisms. One can also reformulate this statement as an equivalence between the categories of Stonean locales and Boolean locales.

Unlike the corresponding statement for Stonean spaces, this version is fully constructive and is valid in any W-topos.

In fact, the traditional Stonean duality? is an immediate consequence of the localic Stonean duality and the spatiality of Stonean locales.

References

The definition is due to Marshall Stone, see

\bibitem{ACSBR} Marshall~H.~Stone, Algebraic characterizations of special Boolean rings. Fundamenta Mathematicae 29:1 (1937), 223–303.

Further investigations of Stonean and hyperstonean? spaces were done by Dixmier:

\bibitem{HS} Jacques Dixmier, Sur certains espaces considérés par M.~H.~Stone. Summa Brasiliensis Mathematicae 2 (1951), 151–182. PDF.

An expository account is available in Takesaki’s book

\bibitem{TOA} Masamichi Takesaki, Theory of operator algebras I. Encyclopaedia of Mathematical Sciences 124 (2002).

Last revised on July 24, 2019 at 00:59:50. See the history of this page for a list of all contributions to it.