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Metrically generated theories


Authors: E. Colebunders and R. Lowen
Journal: Proc. Amer. Math. Soc. 133 (2005), 1547-1556
MSC (2000): Primary 54B30, 18B99, 18E20
DOI: https://doi.org/10.1090/S0002-9939-04-07633-6
Published electronically: November 19, 2004
MathSciNet review: 2111956
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Abstract: Many examples are known of natural functors $K$ describing the transition from categories $\mathcal{C}$ of generalized metric spaces to the ``metrizable" objects in some given topological construct $\mathcal{X}$. If $K$ 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.


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Additional Information

E. Colebunders
Affiliation: Vrije Universiteit Brussel, Vakgroep Wiskunde, Pleinlaan 2, 1050 Brussel, Belgium
Email: evacoleb@vub.ac.be

R. Lowen
Affiliation: Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, 2020 Antwerp, Belgium
Email: bob.lowen@ua.ac.be

DOI: https://doi.org/10.1090/S0002-9939-04-07633-6
Keywords: Topological construct, topological space, metric space, uniform space, approach space, bornological space, measurable space
Received by editor(s): September 22, 2003
Received by editor(s) in revised form: January 5, 2004
Published electronically: November 19, 2004
Communicated by: Alan Dow
Article copyright: © Copyright 2004 American Mathematical Society