Metric spaces in synthetic topology |
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Authors: | Andrej Bauer Davorin Lešnik |
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Institution: | a Department of Mathematics and Physics, University of Ljubljana, Sloveniab Institute for Mathematics, Physics and Mechanics, Ljubljana, Slovenia |
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Abstract: | We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and metric topologies of complete separable metric spaces coincide if they do so for Baire space. In Russian Constructivism the match between synthetic and metric topology breaks down, as even a very simple complete totally bounded space fails to be compact, and its topology is strictly finer than the metric topology. In contrast, in Brouwer’s intuitionism synthetic and metric notions of topology and compactness agree. |
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Keywords: | 26E40 54E35 18B25 |
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