Uniformizable and realcompact bornological universes
Keywords:Bornology, Uniform space, Totally bounded, Realcompactness
AbstractBornological universes were introduced some time ago by Hu and obtained renewed interest in recent articles on convergence in hyperspaces and function spaces and optimization theory. One o fHu's results gives us a necessary and sufficient condition for which a bornological universe is metrizable. In this article we will extend thi sresult and give a characterization of uniformizable bornological universes. Furthermore, a construction on bornological universes that the author used to find the bornological dual of function spaces endowed with the bounded-open topology will be used to define realcompactness for bornological universes. We will also give various characterizations of this new concept.
G. Beer, Embeddings of bornological universes, Set-valued Analysis 16 (2008), 477–488. http://dx.doi.org/10.1007/s11228-007-0068-2
G. Beer, S. Naimpally, J. Rodríguez-López, S-topologies and bounded convergences, Journal of Mathematical Analysis and Applications 339 (2008), 542–552. http://dx.doi.org/10.1016/j.jmaa.2007.07.010
G. Beer, S. Levi, Gap, excess and bornological convergence, Set-Valued Analysis 16 (2008), 489-506. http://dx.doi.org/10.1007/s11228-008-0086-8
G. Beer, S. Levi, Pseudometrizable bornological convergence is Attouch-Wets convergence, Journal of Convex Analysis 15 (2008), 439–453.
G. Beer, S. Levi, Total boundedness and bornologies, Topology and its applications 156 (2009), 1271-1288. http://dx.doi.org/10.1016/j.topol.2008.12.030
G. Beer, M. Segura, Well-posedness, bornologies, and the structure of metric spaces, Applied general topology 10 (2009), 131–157.
G. Beer, S. Levi, Strong uniform continuity, Journal of Mathematical Analysis and Applications 350 (2009), 568–589. http://dx.doi.org/10.1016/j.jmaa.2008.03.058
N. Bourbaki, Topologie Générale, (Hermann, Paris, 1965).
L. Gilman, M. Jerison. Rings of Continuous Functions. (D. Van Nostrand Company, New York, 1960).
J. Hejcman, Boundedness in uniform spaces and topological groups, Czechoslovak Mathematical Journal 84 (1959), 544–563.
S. Hu, Boundedness in a topological space, Journal de Mathématiques Pures et Appliquées 28 (1949).
S. Hu, Introduction to General Topology, (Holden-Day, San Francisco, 1966).
J. Schmets, Espaces de fonctions continues, Lecture Notes in Mathematics 519 (1976).
A. Lechicki, S. Levi, A. Spakowski, Bornological convergences, Journal of Mathematical Analysis and Applications 297 (2004), 751–770. http://dx.doi.org/10.1016/j.jmaa.2004.04.046
T. Vroegrijk, Pointwise bornological spaces, Topology and its Applications 156 (2009), 2019–2027. http://dx.doi.org/10.1016/j.topol.2009.03.040
T. Vroegrijk, Pointwise bornological vector spaces, Topology and its applications, to appear.
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