Comparability of bornological convergences on
the hyperspace of a uniformizable space
Marco Rosa∗
Paolo Vitolo∗
[email protected]
[email protected]
Abstract
Given a compatible uniformity U and an arbitrary bornology S on a topological
space X, we can define on the hyperspace CL(X) of non-empty closed subsets
of X, three convergences related to U and S: the upper, lower and “two-sided”
bornological convergence ([1, 2, 3]). Bornological convergence is a generalization
of the well known Attouch–Wets convergence.
We consider two compatible uniformities U and V, two arbitrary bornologies
S and T on X and we give necessary and sufficient conditions for the comparability of the lower, upper and “two-sided” bornological convergences they
generate.
We also focus on the particular case of the bornology of “Bourbaki bounded”
sets with respect to a uniformity.
Moreover, we consider the bounded-proximal topology generated by an arbitrary bornology and by the proximity induced by a compatible uniformity. We
characterize the comparability of convergences induced by bounded-proximal
topologies related to two compatible uniformities and two arbitrary bornologies.
Keywords: Hyperspace, Attouch-Wets Topology, Bornological Convergence,
Uniform Space, Bounded-proximal Topology.
MSC[2010]: Primary 54B20, Secondary 54E15, 54A20.
References
[1] Gerald Beer, Camillo Costantini, and Sandro Levi. Bornological convergence
and shields. Mediterr. J. Math., 10(1):529–560, 2013.
[2] Gerald Beer and Sandro Levi. Pseudometrizable bornological convergence
is Attouch-Wets convergence. J. Convex Anal., 15(2):439–453, 2008.
[3] A. Lechicki, S. Levi, and A. Spakowski. Bornological convergences. J. Math.
Anal. Appl., 297(2):751–770, 2004. Special issue dedicated to John Horv´ath.
∗ Dipartimento di Matematica, Informatica ed Economia, Universit`
a degli studi della Basilicata, Via dell’Ateneo Lucano 10, 85100 Potenza (Italy)
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