THE BAIRE CLOSURE AND ITS LOGIC

G. BEZHANISHVILI, D. FERNÁNDEZ-DUQUE
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引用次数: 0

Abstract

The Baire algebra of a topological space X is the quotient of the algebra of all subsets of X modulo the meager sets. We show that this Boolean algebra can be endowed with a natural closure operator, resulting in a closure algebra which we denote Abstract Image$\mathbf {Baire}(X)$. We identify the modal logic of such algebras to be the well-known system Abstract Image$\mathsf {S5}$, and prove soundness and strong completeness for the cases where X is crowded and either completely metrizable and continuum-sized or locally compact Hausdorff. We also show that every extension of Abstract Image$\mathsf {S5}$ is the modal logic of a subalgebra of Abstract Image$\mathbf {Baire}(X)$, and that soundness and strong completeness also holds in the language with the universal modality.

贝叶封闭及其逻辑
拓扑空间 X 的贝叶尔代数是 X 的所有子集调制集代数的商。我们证明,这个布尔代数可以被赋予一个自然闭包算子,从而得到一个闭包代数,我们将其命名为 $\mathbf {Baire}(X)$ 。我们确定这种代数的模态逻辑是著名的 $\mathsf {S5}$ 系统,并证明了 X 是拥挤的、完全可元化的和连续体大小的或局部紧凑的 Hausdorff 的情况下的健全性和强完备性。我们还证明$edmathsf {S5}$的每一个扩展都是$\mathbf {Baire}(X)$ 的一个子代数的模态逻辑,并且在具有普遍模态的语言中健全性和强完备性也成立。
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