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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) 1
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