Abstract

This paper is devoted to study the following question: given a bornology B\mathbf{B} on a metric space (X,d), when is there a metric d' on X, uniformly equivalent to d, for which the bornology of d'-bounded sets coincides with B\mathbf{B}? If that happens, we will say that B\mathbf{B} is uniformly metrizable. We characterize these kinds of bornologies on X, and we obtain necessary and sufficient conditions in order to some of the most common bornologies enjoy this property. More precisely, we will consider the compact bornology, the totally bounded bornology and the bornology of the Bourbaki-bounded sets.

Author information

Contact details are reproduced from the original publication and may be historical.

Isabel Garrido

Dep. de Geometría y Topología, Universidad Complutense de Madrid, 28040 Madrid, Spain

maigarri@mat.ucm.es

Ana S. Meroño

Dep. de Geometría y Topología, Universidad Complutense de Madrid, 28040 Madrid, Spain

anamerogno@gmail.com

Keywords

  • Metric bounded sets
  • bornological universe
  • totally-bounded sets
  • Bourbaki-bounded sets
  • uniform equivalent metrics

Mathematics Subject Classification

54E35, 54E45; 46A17

References

  1. M. Atsuji: Uniform continuity of continuous functions of metric spaces, Pac. J. Math. 8 (1958) 11–16.https://doi.org/10.2140/pjm.1958.8.11Google Scholar
  2. N. Bourbaki: Topologie Générale, Hermann, Paris (1965).Google Scholar
  3. G. Beer: On metric boundedness structures, Set-Valued Anal. 7 (1999) 195–208.https://doi.org/10.1023/A:1008720619545Google Scholar
  4. G. Beer, C. Constantini, S. Levi: Total boundedness in metrizable spaces, Houston J. Math. 37 (2011) 1347–1362.Google Scholar
  5. G. Beer, S. Levi: Gap, excess and bornological convergence, Set-Valued Anal. 16 (2008) 489–506.https://doi.org/10.1007/s11228-008-0086-8Google Scholar
  6. R. Engelking: General Topology, Heldermann, Berlin (1989).Google Scholar
  7. M. I. Garrido, J. A. Jaramillo: Lipschitz-type functions on metric spaces, J. Math. Anal. Appl. 380 (2008) 282–290.https://doi.org/10.1016/j.jmaa.2007.08.028Google Scholar
  8. J. L. Kelley: General Topology, Springer, New York (1975).Google Scholar
  9. J. Hejcman: Boundedness in uniform spaces and topological groups, Czech. Math. J. 9 (1959) 544–561.https://doi.org/10.21136/CMJ.1959.100381Google Scholar
  10. J. Hejcman: On conservative uniform spaces, Commentat. Math. Univ. Carol. 7 (1966) 411–417.Google Scholar
  11. J. Hejcman: On simple recognizing of bounded sets, Commentat. Math. Univ. Carol. 38 (1997) 149–156.Google Scholar
  12. S. T. Hu: Boundedness in a topological space, J. Math. Pures Appl. 228 (1949) 287–320.Google Scholar
  13. M. D. Potter: The uniform structure of boundedly compact spaces, Bull. Aust. Math Soc. 38 (1988) 95–97.https://doi.org/10.1017/S0004972700027295Google Scholar
  14. H. E. Vaughan: On locally compact metrisable spaces, Bull. Amer. Math. Soc. 43 (1937) 532–535.https://doi.org/10.1090/S0002-9904-1937-06593-1Google Scholar
  15. T. Vroegrijk: Uniformizable and realcompact bornological universes, Appl. Gen. Topol. 10 (2009) 277–287.https://doi.org/10.4995/agt.2009.1740Google Scholar
  16. R. Williamson, L. Janos: Constructing metrics with the Heine-Borel property, Proc. Amer. Math. Soc. 100 (1987) 567–573.https://doi.org/10.1090/S0002-9939-1987-0891165-XGoogle Scholar

Suggested citation

I. Garrido, A. S. Meroño. “Uniformly Metrizable Bornologies.” Journal of Convex Analysis 20 (2013), No. 1, 285–299. https://doi.org/10.68381/jca20018

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.