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Overlap Algebras as Almost Discrete Locales

Francesco Ciraulo
arXiv: Logic
Jan 2016
摘要
Boolean locales are discrete, in the sense that a spatial Boolean locale is the same thing as the frame of open subsets of a discrete space, that is, the powerset of a set. This basic fact, however, cannot be proven constructively, that is, over intuitionistic logic. Actually, Boolean locales are almost discrete if and only if the law of excluded middle (LEM) holds. In fact, discrete locales are never Boolean constructively, except for the trivial locale. So, what is an almost discrete locale constructively? Our claim is that Sambin's Overlap Algebras have good enough features to deserve to be called that. For instance, they include the class of discrete locales, they arise as smallest strongly dense sublocales (of overt locales), and hence they coincide with the Boolean locales if LEM holds.
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