Apartness and Uniformity - Preamble
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منابع مشابه
Precompact Apartness Spaces
We present a notion of precompactness, and study some of its properties, in the context of apartness spaces whose apartness structure is not necessarily induced by any uniform one. The presentation lies entirely with a Bishop-style constructive framework, and is a contribution to the ongoing development of the constructive theories of apartness and uniformity.
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The theory of apartness spaces, and their relation to topological spaces (in the point—set case) and uniform spaces (in the set—set case), is sketched. New notions of local decomposability and regularity are investigated, and the latter is used to produce an example of a classically metrisable apartness on R that cannot be induced constructively by even a uniform structure.
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Apartness spaces were introduced as a constructive counterpart to proximity spaces which, in turn, aimed to model the concept of nearness of sets in a metric or topological environment. In this paper we introduce apartness algebras and apartness frames intended to be abstract counterparts to the apartness spaces of Bridges and Vîta [3], and we prove a discrete duality for them.
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On more than one occasion, the late Errett Bishop [1] suggested that general pointset topology might have little signi cance for constructive mathematics, and that locally convex spaces, together with appropriate ad hoc topological constructions, might su¢ ce for most constructive analysis. Work begun by us ve years ago suggests that, on the contrary, there is at least one constructive approa...
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