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Sets completely separated by functions in Bishop Set Theory

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arxiv 2208.07826 v1 pith:74NDR5JE submitted 2022-08-16 math.LO

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keywords completelyseparatedsetsspacesbishopgiventheoryfunctions
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Within Bishop Set Theory, a reconstruction of Bishop's theory of sets, we study the so-called completely separated sets, that is sets equipped with a positive notion of an inequality, induced by a given set of real-valued functions. We introduce the notion of a global family of completely separated sets over an index-completely separated set, and we describe its Sigma- and Pi-set. The free completely separated set on a given set is also presented. Purely set-theoretic versions of the classical Stone-\v{C}ech theorem and the Tychonoff embedding theorem for completely regular spaces are given, replacing topological spaces with function spaces and completely regular spaces with completely separated sets.

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