Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Uniform continuity on bounded sets and the Attouch-Wets topology

Authors: Gerald Beer and Anna Di Concilio
Journal: Proc. Amer. Math. Soc. 112 (1991), 235-243
MSC: Primary 54B20
MathSciNet review: 1033956
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\text{CL(}}X{\text{)}}$ be the nonempty closed subsets of a metrizable space $ X$. If $ d$ is a compatible metric, the metrizable Attouch-Wets topology $ {\tau _{aw}}(d)$ on $ {\text{CL(}}X{\text{)}}$ is the topology of uniform convergence of distance functionals associated with elements of $ {\text{CL(}}X{\text{)}}$ 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.

References [Enhancements On Off] (What's this?)

  • [At] H. Attouch, Variational convergence for functions and operators, Pitman, Boston, 1984. MR 773850 (86f:49002)
  • [ALW] H. Attouch, R. Lucchetti, and R. Wets, The topology of the $ \rho $-Hausdorff distance, Ann. Mat. Pura Appl. (to appear).
  • [AW] H. Attouch and R. Wets, Quantitative stability of variational systems. I. The epigraphical distance, Working paper, IIASA, Laxenburg, Austria, 1988.
  • [Az] D. Azé, On some metric aspects of set convergence, preprint.
  • [AP1] D. Azé and J.-P. Penot, Operations on convergent families of sets and functions, AVAMAC report, Perpignan, 1987.
  • [AP2] -, Recent quantitative results about the convergence of convex sets and functions, in Functional Analysis and Approximation (P. L. Papini, ed.), Pitagora Editrice, Bologna, 1989. MR 1001572 (90k:90120)
  • [Be1] G. Beer, Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance, Proc. Amer. Math. Soc. 95 (1985), 653-658. MR 810180 (87e:54024)
  • [Be2] -, Metric spaces with nice closed balls and distance functions for closed sets, Bull. Australian Math. Soc. 35 (1987), 81-96. MR 875510 (88e:54033)
  • [Be3] -, On Mosco convergence of convex sets, Bull. Austral. Math. Soc. 38 (1988), 239-253. MR 969914 (90a:46026)
  • [Be4] -, On the Young-Fenchel transform for convex functions, Proc. Amer. Math. Soc. 104 (1988), 1115-1123. MR 937844 (89i:49006)
  • [Be5] -, Convergence of continuous linear functionals and their level sets, Arch. Math. 52 (1989), 482-491. MR 998621 (90i:46018)
  • [Be6] -, Conjugate convex functions and the epi-distance topology, Proc. Amer. Math. Soc. 108 (1990), 117-126. MR 982400 (90f:46018)
  • [BB] G. Beer and J. Borwein, Mosco convergence and reflexivity, Proc. Amer. Math. Soc. 109 (1990), 427-436. MR 1012924 (91c:46016)
  • [BL] G. Beer and R. Lucchetti, Convex optimization and the epi-distance topology, Trans. Amer. Math. Soc. (to appear). MR 1012526 (92a:49018)
  • [BF] J. Borwein and S. Fitzpatrick, Mosco convergence and the Kadec property, Proc. Amer. Math. Soc. 106 (1989), 843-849. MR 969313 (90i:46025)
  • [CV] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Math., vol. 580, Springer-Verlag, Berlin, 1975. MR 0467310 (57:7169)
  • [CP] L. Contesse and J.-P. Penot, Continuity of polarity and conjugacyfor the epi-distance topology, preprint.
  • [FLL] S. Francaviglia, A. Lechicki, and S. Levi, Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl. 112 (1985), 347-370. MR 813603 (87e:54025)
  • [Ku] K. Kuratowski, Topology, vol. 1, Academic Press, New York, 1966. MR 0217751 (36:840)
  • [Mi] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. MR 0042109 (13:54f)
  • [Mo1] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. in Math. 3 (1969), 510-585. MR 0298508 (45:7560)
  • [Mo2] -, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971), 518-535. MR 0283586 (44:817)
  • [Na] S. Naimpally, Graph topology for function spaces, Trans. Amer. Math. Soc. 123 (1966), 267-272. MR 0192466 (33:691)
  • [NW] S. Naimpally and B. Warrack, Proximity spaces, Cambridge University Press, Cambridge, 1970. MR 0278261 (43:3992)
  • [SW] G. Salinetti and R. Wets, On the relation between two types of convergence for convex functions, J. Math. Anal. Appl. 60 (1977), 211-226. MR 0479398 (57:18828)
  • [So] 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.
  • [TL] A. Taylor and D. Lay, Introduction to functional analysis, 2nd ed., Wiley, New York, 1980. MR 564653 (81b:46001)
  • [Ts] M. Tsukada, Convergence of best approximations in a smooth Banach space, J. Approx. Theory 40 (1984), 301-309. MR 740641 (86a:41034)
  • [WW] D. Walkup and R. Wets, Convergence of some convex-cone valued mappings, Proc. Amer. Math. Soc. 18 (1967), 229-235. MR 0209806 (35:702)
  • [Wi] R. Wijsman, Convergence of sequences of convex sets, cones, and functions. II, Trans. Amer. Math. Soc. 123 (1966), 32-45. MR 0196599 (33:4786)
  • [Wl] S. Willard, General topology, Addison-Wesley, Reading, MA, 1968. MR 0264581 (41:9173)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54B20

Retrieve articles in all journals with MSC: 54B20

Additional Information

Keywords: Attouch-Wets topology, uniform continuity on bounded sets, uniform convergence of distance functionals on bounded sets, set convergence, hyperspace
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society