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$\operatorname{UC}$ space; boundedly $\operatorname{UC}$ space; boundedly compact space; compact-open to\-po\-logy; topology of uniform convergence on bounded sets; Attouch-Wets topology; bounded proximal topology
Following the paper [BDC1], further relations between the classical topologies on function spaces and new ones induced by hyperspace topologies on graphs of functions are introduced and further characterizations of boundedly $\operatorname{UC}$ spaces are given.
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[AW] Attouch H., Wets R.: Quantitative stability of variational systems I, The epigraphical distances. preprint. MR 1018570
[ALW] Attouch H., Lucchetti R., Wets R.: The topology of the $\rho $-Hausdorff distance. to appear in Annali Mat. Pure Appl. Zbl 0769.54009
[AP] Aze D., Penot J.: Operations on convergent families of sets and functions. Optimization 21 (1990), 521-534. MR 1069660 | Zbl 0719.49013
[Be] Beer G.: Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance. Proc. Amer. Math. Soc. 95 (1985), 653-658. MR 0810180 | Zbl 0594.54007
[BDC1] Beer G., DiConcilio A.: A generalization of boundedly compact metric spaces. Comment. Math. Univ. Carolinae 32 (1991), 362-367. MR 1137797
[BDC2] Beer G., DiConcilio A.: Uniform continuity on bounded sets and the Attouch-Wets topology. Proc. Amer. Math. Soc. 112 (1991), 235-244. MR 1033956
[BL1] Beer G., Lucchetti R.: Weak topologies for the closed subsets of a metrizable space. to appear in Trans. Amer. Math. Soc. MR 1094552 | Zbl 0810.54011
[BL2] Beer G., Lucchetti R.: Well-posed optimization problems and a new topology for the closed subsets of a metric space. preprint. MR 1256444 | Zbl 0812.54015
[BLLN] Beer G., Lechicki A., Levi S., Naimpally S.: Distance functionals and the suprema of hyperspace topologies. to appear in Annali Mat. Pure Appl.
[Ho] Holá Ł.: The Attouch-Wets topology and a characterization of normable linear spaces. Bull. Austral. Math. Soc. 44 (1991), 11-18. MR 1120389
[LL] Lechicki A., Levi S.: Wijsman convergence in the hyperspace of a metric space. Bull. Un. Mat. Ital. 5-B (1987), 435-452. MR 0896334 | Zbl 0655.54007
[Pe] Penot J.P.: The cosmic Hausdorff topology, the bounded Hausdorff topology, and continuity of polarity. to appear in Proc. Amer. Math. Soc. MR 1068129 | Zbl 0774.54008
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