  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.
