Apartness, Topology, and Uniformity: a Constructive View
نویسندگان
چکیده
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.
منابع مشابه
Journey into Apartness Space
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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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 formal spaces and the more recent theory of apartness spaces have a priori not much more in common than that each of them was initiated as a constructive approach to general topology. We nonetheless try to do the first steps in relating these competing theories to each other. Formal topology was put forward in the mid 1980s by Sambin [8] in order to make available to Martin–Löf’s ...
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ورودعنوان ژورنال:
- Math. Log. Q.
دوره 48 شماره
صفحات -
تاریخ انتشار 2002