Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Metrically generated theories

Authors: E. Colebunders and R. Lowen
Journal: Proc. Amer. Math. Soc. 133 (2005), 1547-1556
MSC (2000): Primary 54B30, 18B99, 18E20
Published electronically: November 19, 2004
MathSciNet review: 2111956
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. J. Adámek, H. Herrlich and G.E. Strecker. Abstract and Concrete Categories. John Wiley & Sons, New York, 1990. MR 1051419 (91h:18001)
  • 2. G. Choquet. Convergences. Ann. Univ. Grenoble, Sect. Math. Phys., 23 (1948), 57-112. MR 0025716 (10,53d)
  • 3. E. Colebunders, R. Lowen. Topological quasitopos hulls of categories containing topological and metric objects, Cahiers Topol. Géom. Diff. Catég. 30 (1989), 213-228. MR 1029625 (91a:54015)
  • 4. E. Colebunders, R. Lowen and M. Sioen. Saturated collections of metrics, preprint.
  • 5. D. Dikranjan, W. Tholen. Categorical Structure of Closure Operators, Mathematics and its Applications, Kluwer Academic Publishers, 1995. MR 1368854 (97i:18004)
  • 6. A. Frölicher and A. Kriegl. Linear Spaces and Differentiation Theory, Pure and Applied Mathematics, John Wiley & Sons, 1988. MR 0961256 (90h:46076)
  • 7. L. Gillman, M. Jerison. Rings of Continuous Functions, Springer Verlag, 1976. MR 0407579 (53:11352)
  • 8. H. Herrlich. A concept of nearness, Gen. Top. Appl., 4 (1974), 191-212. MR 0350701 (50:3193)
  • 9. H.-P. Künzi. Nonsymmetric distances and their associated topologies: about the origins of basic ideas in the area of asymmetric topology, Handbook of the History of General Topology, Volume 3, Kluwer Academic Publishers, (2001), 853-968. MR 1900267 (2003d:54001)
  • 10. R. Lowen. Approach Spaces: the Missing Link in the Topology-Uniformity-Metric Triad, Oxford Mathematical Monographs, Oxford University Press, 1997. MR 1472024 (98f:54002)
  • 11. S.A. Naimpally, B.D. Warrack. Proximity Spaces, Cambridge University Press, 1970. MR 0278261 (43:3992)
  • 12. G. Preuss. Theory of Topological Structures, Mathematics and its Applications, Kluwer Academic Publishers, 1988. MR 0937052 (89m:54014)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 54B30, 18B99, 18E20

Retrieve articles in all journals with MSC (2000): 54B30, 18B99, 18E20

Additional Information

E. Colebunders
Affiliation: Vrije Universiteit Brussel, Vakgroep Wiskunde, Pleinlaan 2, 1050 Brussel, Belgium

R. Lowen
Affiliation: Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, 2020 Antwerp, Belgium

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

American Mathematical Society