Choice-free Stone duality

TL;DR

This paper presents a choice-free topological representation and duality for Boolean algebras using spectral spaces and regular open sets, avoiding nonconstructive principles like the Boolean Prime Ideal Theorem.

Contribution

It introduces a novel choice-free representation of Boolean algebras via spectral spaces and establishes a dual equivalence with a subcategory of spectral spaces, connecting Stone and Tarski's ideas.

Findings

Provides a choice-free representation of Boolean algebras.

Establishes a dual equivalence between Boolean algebras and spectral spaces.

Connects Stone and Tarski's approaches through Vietoris hyperspaces.

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 paper, 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. 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.