Choice-free Stone duality
Nick Bezhanishvili, Wesley H. Holliday
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 topo-
logical 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 �eld of sets; but like Stone's repre-
sentation, it provides the bene�t 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 homomor-
phisms 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.