Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Weak topologies for the closed subsets of a metrizable space

Authors: Gerald Beer and Roberto Lucchetti
Journal: Trans. Amer. Math. Soc. 335 (1993), 805-822
MSC: Primary 54B20; Secondary 54E35
MathSciNet review: 1094552
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this article is to propose a unified theory for topologies on the closed subsets of a metrizable space. It can be shown that all of the standard hyperspace topologies--including the Hausdorff metric topology, the Vietoris topology, the Attouch-Wets topology, the Fell topology, the locally finite topology, and the topology of Mosco convergence--arise as weak topologies generated by families of geometric functionals defined on closed sets. A key ingredient is the simple yet beautiful interplay between topologies determined by families of gap functionals and those determined by families of Hausdorff excess functionals.

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

  • [At] H. Attouch, Variational convergence for functions and operators, Pitman, New York, 1984. MR 773850 (86f:49002)
  • [AW] H. Attouch and R. Wets, Quantitative stability of variational systems. I. The epigraphical distance, Trans. Amer. Math. Soc. 328 (1991), 695-729. MR 1018570 (92c:90111)
  • [AAB] H. Attouch, D. Azé, and G. Beer, On some inverse stability problems for the epigraphical sum, Nonlinear Anal. 16 (1991), 241-254. MR 1091522 (92b:49021)
  • [ALW] H. Attouch, R. Lucchetti, and R. Wets, The topology of the $ \rho $-Hausdorff distance, Ann. Mat. Pura Appl. (to appear).
  • [AP] D. Azé and J.-P. Penot, Operations on convergent families of sets and functions, Optimization 21 (1990), 521-534. MR 1069660 (92b:49022)
  • [Be1] G. Beer, On Mosco convergence of convex sets, Bull. Austral. Math. Soc. 38 (1988), 239-253. MR 969914 (90a:46026)
  • [Be2] -, Convergence of continuous linear functionals and their level sets, Arch. Math. 52 (1989), 482-491. MR 998621 (90i:46018)
  • [Be3] -, Conjugate convex functions and the epi-distance topology, Proc. Amer. Math. Soc. 108 (1990), 117-126. MR 982400 (90f:46018)
  • [Be4] -, An embedding theorem for the Fell topology, Michigan Math. J. 35 (1988), 239-253. MR 931935 (89e:54017)
  • [Be5] -, Support and distance functionals for convex sets, Numer. Funct. Anal. Optim. 10 (1989), 15-36. MR 978800 (89m:46031)
  • [Be6] -, A Polish topology for the closed subsets of a Polish space, Proc. Amer. Math. Soc. 113 (1991), 1123-1133. MR 1065940 (92c:54009)
  • [Be7] -, Mosco convergence and weak topologies for convex sets and functions, Mathematika 38 (1991), 89-104. MR 1116688 (92j:46022)
  • [BB] G. Beer and J. Borwein, Mosco convergence and reflexivity, Proc. Amer. Math. Soc. 109 (1990), 427-436. MR 1012924 (91c:46016)
  • [BDC] G. Beer and A. Di Concilio, Uniform continuity on bounded sets and the Attouch- Wets topology, Proc. Amer. Math. Soc. 112 (1991), 235-243. MR 1033956 (91h:54013)
  • [BHPV] G. Beer, C. Himmelberg, K. Prikry, and F. Van Vleck, The locally finite topology on $ {2^X}$, Proc. Amer. Math. Soc. 101 (1987), 168-172. MR 897090 (88f:54014)
  • [BLLN] G. Beer, A. Lechicki, S. Levi, and S. Naimpally, Distance functionals and the suprema of hyperspace topologies, Ann. Mat. Pura Appl. (to appear). MR 1199663 (94c:54016)
  • [BL1] G. Beer and R. Lucchetti, Convex optimization and the epi-distance topology, Trans. Amer. Math. Soc. 327 (1991), 795-813. MR 1012526 (92a:49018)
  • [BL2] -, Well-posed optimization problems and a new topology for the closed subsets of a metrizable space, Rocky Mountain J. Math., (to appear).
  • [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 and New York, 1977. MR 0467310 (57:7169)
  • [Co] B. Cornet, Topologies sur les fermés d'un espace métrique, Cahiers de mathématiques de la décision, vol. 7309, Université de Paris Dauphine, Paris, 1973.
  • [En] R. Engleking, General topology, Polish Scientific Publishers, Warsaw, 1977. MR 0500780 (58:18316b)
  • [Fe] J. Fell, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc. 13 (1962), 472-476. MR 0139135 (25:2573)
  • [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)
  • [KT] E. Klein and A. Thompson, Theory of correspondences, Wiley, New York, 1984. MR 752692 (86a:90012)
  • [LL] A. Lechicki and S. Levi, Wijsman convergence in the hyperspace of a metric space, Bull. Un. Mat. Ital. 5-B (1987), 435-452. MR 896334 (88e:54007)
  • [Mi] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. MR 0042109 (13:54f)
  • [Mo] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. in Math. 3 (1969), 510-585. MR 0298508 (45:7560)
  • [NS] S. Naimpally and P. Sharma, Fine uniformity and the locally finite hyperspace topology on $ {2^X}$, Proc. Amer. Math. Soc. 103 (1988), 641-646. MR 943098 (89f:54021)
  • [Pe] J.-P. Penot, The cosmic Hausdorff topology, the bounded Hausdorff topology, and continuity of polarity, Proc. Amer. Math. Soc. 113 (1991), 275-285. MR 1068129 (91k:54012)
  • [SP] P. Shunmugaraj and D. V. Pai, On stability of approximate solutions of minimization problems, Numer. Funct. Anal. Optim. (to appear). MR 1159932 (93e:49042)
  • [SZ1] Y. Sonntag and C. Zalinescu, Scalar convergence of convex sets, J. Math. Anal. Appl. 164 (1992), 219-241. MR 1146585 (93c:52002)
  • [SZ2] -, Set convergences. An attempt of classification, Proc. Internat. Conf. on Differential Equations and Control Theory (Iasi, Romania, August, 1990), Pitman Research Notes in Math. #250, 1991, pp. 312-323.
  • [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)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54B20, 54E35

Retrieve articles in all journals with MSC: 54B20, 54E35

Additional Information

Keywords: Hyperspace, gap functional, Hausdorff excess, Hausdorff metric topology, Attouch-Wets topology, Wijsman topology, Vietoris topology, distance functional, weak topologies
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society