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Uniform continuity on bounded sets and the Attouch-Wets topology. (English) Zbl 0677.54007

Let CL(X) be the nonempty closed subses of a metrizable space X. If d is a compatible metric, the metrizable Attouch-Wets topology \(\tau_{aw}(d)\) on CL(X) is the topology of uniform convergence of distance functionals associated with elements of CL(X) on bounded subsets of X. The main result of this paper shows that two compatible metrics d and \(\rho\) determine the same Attouch-Wets topologies if and only if they determine the same bounded sets and the same class of functions that are uniformly continuous on bounded sets.
Reviewer: G.Beer

MSC:

54B20 Hyperspaces in general topology
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
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[1] H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. · Zbl 0561.49012
[2] H. Attouch, R. Lucchetti, and R. Wets, The topology of the \( \rho \)-Hausdorff distance, Ann. Mat. Pura Appl. (to appear). · Zbl 0769.54009
[3] H. Attouch and R. Wets, Quantitative stability of variational systems. I. The epigraphical distance, Working paper, IIASA, Laxenburg, Austria, 1988. · Zbl 0753.49007
[4] D. Azé, On some metric aspects of set convergence, preprint.
[5] D. Azé and J.-P. Penot, Operations on convergent families of sets and functions, AVAMAC report, Perpignan, 1987. · Zbl 0719.49013
[6] Dominique Azé and Jean-Paul Penot, Recent quantitative results about the convergence of convex sets and functions, Functional analysis and approximation (Bagni di Lucca, 1988) Pitagora, Bologna, 1989, pp. 90 – 110.
[7] Gerald Beer, Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance, Proc. Amer. Math. Soc. 95 (1985), no. 4, 653 – 658. · Zbl 0594.54007
[8] Gerald Beer, Metric spaces with nice closed balls and distance functions for closed sets, Bull. Austral. Math. Soc. 35 (1987), no. 1, 81 – 96. · Zbl 0588.54014 · doi:10.1017/S000497270001306X
[9] Gerald Beer, On Mosco convergence of convex sets, Bull. Austral. Math. Soc. 38 (1988), no. 2, 239 – 253. · Zbl 0669.52002 · doi:10.1017/S0004972700027519
[10] Gerald Beer, On the Young-Fenchel transform for convex functions, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1115 – 1123. · Zbl 0691.46006
[11] Gerald Beer, Convergence of continuous linear functionals and their level sets, Arch. Math. (Basel) 52 (1989), no. 5, 482 – 491. · Zbl 0662.46015 · doi:10.1007/BF01198356
[12] Gerald Beer, Conjugate convex functions and the epi-distance topology, Proc. Amer. Math. Soc. 108 (1990), no. 1, 117 – 126. · Zbl 0681.46014
[13] Gerald Beer and Jonathan M. Borwein, Mosco convergence and reflexivity, Proc. Amer. Math. Soc. 109 (1990), no. 2, 427 – 436. · Zbl 0763.46006
[14] Gerald Beer and Roberto Lucchetti, Convex optimization and the epi-distance topology, Trans. Amer. Math. Soc. 327 (1991), no. 2, 795 – 813. · Zbl 0681.46013
[15] Jonathan M. Borwein and Simon Fitzpatrick, Mosco convergence and the Kadec property, Proc. Amer. Math. Soc. 106 (1989), no. 3, 843 – 851. · Zbl 0672.46007
[16] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. · Zbl 0346.46038
[17] L. Contesse and J.-P. Penot, Continuity of polarity and conjugacyfor the epi-distance topology, preprint. · Zbl 0772.46044
[18] Sebastiano Francaviglia, Alojzy Lechicki, and Sandro Levi, Quasiuniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl. 112 (1985), no. 2, 347 – 370. · Zbl 0587.54003 · doi:10.1016/0022-247X(85)90246-X
[19] K. Kuratowski, Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. · Zbl 0158.40901
[20] Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152 – 182. · Zbl 0043.37902
[21] Umberto Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math. 3 (1969), 510 – 585. · Zbl 0192.49101 · doi:10.1016/0001-8708(69)90009-7
[22] Umberto Mosco, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971), 518 – 535. · Zbl 0253.46086 · doi:10.1016/0022-247X(71)90200-9
[23] Somashekhar Amrith Naimpally, Graph topology for function spaces, Trans. Amer. Math. Soc. 123 (1966), 267 – 272. · Zbl 0151.29703
[24] S. A. Naimpally and B. D. Warrack, Proximity spaces, Cambridge Tracts in Mathematics and Mathematical Physics, No. 59, Cambridge University Press, London-New York, 1970. · Zbl 0206.24601
[25] Gabriella Salinetti and Roger J.-B. Wets, On the relations between two types of convergence for convex functions, J. Math. Anal. Appl. 60 (1977), no. 1, 211 – 226. · Zbl 0359.54005 · doi:10.1016/0022-247X(77)90060-9
[26] Y. Sonntag, Convergence au sens de Mosco; théorie et applications à l’approximation des solutions d’inéquations, Thèse d’Etat. Université de Provence, Marseille, 1982.
[27] Angus Ellis Taylor and David C. Lay, Introduction to functional analysis, 2nd ed., John Wiley & Sons, New York-Chichester-Brisbane, 1980. · Zbl 0501.46003
[28] Makoto Tsukada, Convergence of best approximations in a smooth Banach space, J. Approx. Theory 40 (1984), no. 4, 301 – 309. · Zbl 0545.41042 · doi:10.1016/0021-9045(84)90003-0
[29] David W. Walkup and Roger J.-B. Wets, Continuity of some convex-cone-valued mappings, Proc. Amer. Math. Soc. 18 (1967), 229 – 235. · Zbl 0145.38004
[30] R. A. Wijsman, Convergence of sequences of convex sets, cones and functions. II, Trans. Amer. Math. Soc. 123 (1966), 32 – 45. · Zbl 0146.18204
[31] Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. · Zbl 1052.54001
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