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Metrically generated theories

Authors:	E Colebunders R Lowen

Institution:	Vrije Universiteit Brussel, Vakgroep Wiskunde, Pleinlaan 2, 1050 Brussel, Belgium ; Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, 2020 Antwerp, Belgium

Abstract:	Many examples are known of natural functors describing the transition from categories $\mathcal{C}$ of generalized metric spaces to the ``metrizable" objects in some given topological construct $\mathcal{X}$ . If preserves initial morphisms and if $K(\mathcal{C})$ is initially dense in $\mathcal{X}$ , then we say that $\mathcal{X}$ is $\mathcal{C}$ -metrically generated. Our main theorem proves that $\mathcal{X}$ is $\mathcal{C}$ -metrically generated if and only if $\mathcal{X}$ can be isomorphically described as a concretely coreflective subconstruct of a model category with objects sets structured by collections of generalized metrics in $\mathcal{C}$ and natural morphisms. This theorem allows for a unifying treatment of many well-known and varied theories. Moreover, via suitable comparison functors, the various relationships between these theories are studied.

Keywords:	Topological construct topological space metric space uniform space approach space bornological space measurable space

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