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.

中文翻译：

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无选择的石头二重性

通过斯通空间的闭开集的布尔代数的标准拓扑表示需要非建设性选择原则，相当于布尔素理想定理。在本文中，我们描述了布尔代数的无选择拓扑表示。这种表示使用了斯通在通过紧开集表示分布格时使用的光谱空间的一个子类。它还利用了 Tarski 的观察，即任何拓扑空间的规则开集形成布尔代数。我们无需选择原则就证明任何布尔代数都来自一个特殊的谱空间X通过紧凑的常规开集X; 这些集合也可以描述为紧凑开放的集合X并且在专业化阶的扰乱拓扑中规则开路X, 允许将一个任意的布尔代数应用于一个关于分离式正则开的简单推理。因此，我们的表示是 Stone 和 Tarski 的混合，两者通过 Vietoris 连接：相关的光谱空间也作为具有上 Vietoris 拓扑的 Stone 空间的非空闭集的超空间出现。这种联系清楚地表明了我们的无选择 Stone 对偶的点集拓扑方法（可称为超空间方法）与使用 Stone 语言环境的无选择 Stone 对偶的无点方法之间的关系。与斯通通过斯通空间对布尔代数的表示不同，我们对布尔代数的无选择拓扑表示并没有表明每个布尔代数都可以表示为一个集合域。但就像斯通的代表一样，它提供了布尔代数的拓扑观点的好处，只是现在别无选择。除了表示之外，我们在具有布尔同态的布尔代数类别与具有谱图的谱空间类别的子类别之间建立了无选择对偶等价。我们展示了如何使用这种对偶性来证明有关布尔代数的一些基本事实。