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CHOICE-FREE STONE DUALITY

Published online by Cambridge University Press:  29 August 2019

NICK BEZHANISHVILI  and
WESLEY H. HOLLIDAY
NICK BEZHANISHVILI
Affiliation:
INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAM1090GE AMSTERDAM, THE NETHERLANDS E-mail:n.bezhanishvili@uva.nl
WESLEY H. HOLLIDAY
Affiliation:
DEPARTMENT OF PHILOSOPHY AND GROUP IN LOGIC AND THE METHODOLOGY OF SCIENCE UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA94720, USA E-mail:wesholliday@berkeley.edu
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Abstract

The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.

Keywords

03G0506E1506D2203E25Stone dualityBoolean algebraregular open algebraVietoris hyperspacespectral spacesStone localesaxiom of choice
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 1 , March 2020 , pp. 109 - 148
Copyright
Copyright © The Association for Symbolic Logic 2019 

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