Abstract

Let S\mathcal{S} be an ideal of subsets of a metric space ⟨X,d⟩\langle X,d \rangle. A net of subsets ⟨Aλ⟩\langle A_\lambda\rangle of XX is called S\mathcal{S}-convergent to a subset AA of XX if for each S∈SS \in \mathcal{S} and each ε>0\varepsilon > 0, we have eventually A∩S⊆Aλε and Aλ∩S⊆Aε.A \cap S \subseteq A^\varepsilon_\lambda \ \textrm{and} \ A_\lambda \cap S \subseteq A^\varepsilon. We identify necessary and sufficient conditions for this convergence to be admissible and topological on the power set of XX. We show that S\mathcal{S}-convergence is compatible with a pseudometrizable topology if and only if S\mathcal{S} has a countable base and each member of S\mathcal{S} has an ε\varepsilon-enlargement that is again in S\mathcal{S}. Further, in the case that the ideal is a bornology, we show that S\mathcal{S}-convergence when pseudometrizable is Attouch-Wets convergence with respect to an equivalent metric.

Author information

Contact details are reproduced from the original publication and may be historical.

Gerald Beer

Dept. of Mathematics, California State University, 5151 State University Drive, Los Angeles, CA 90032, U.S.A.

gbeer@cslanet.calstatela.edu

Sandro Levi

Dip. di Matematica e Applicazioni, Università di Milano-Bicocca, Via Cozzi 53, 20125 Milano, Italy

sandro.levi@unimib.it

Keywords

  • Bornological convergence
  • Attouch-Wets convergence
  • bounded Hausdorff convergence
  • hyperspace
  • bornology

Mathematics Subject Classification

54B20; 46A17, 54E35

References

  1. H. Attouch: Variational Convergence for Functions and Operators, Pitman, Boston (1984).Google Scholar
  2. H. Attouch, R. Lucchetti, R. Wets: The topology of the ρ-Hausdorff distance, Ann. Mat. Pura Appl. 160 (1991) 303–320.https://doi.org/10.1007/BF01764131Google Scholar
  3. H. Attouch, R. Wets: Quantitative stability of variational systems. I: the epigraphical distance, Trans. Amer. Math. Soc. 328 (1991) 695–730.https://doi.org/10.2307/2001800Google Scholar
  4. D. Azé, J.-P. Penot: Operations on convergent families of sets and functions, Optimization 21 (1990) 521–534.https://doi.org/10.1080/02331939008843576Google Scholar
  5. G. Beer: Convergence of linear functionals and their level sets, Arch. Math. 55 (1990) 285–292.https://doi.org/10.1007/BF01191170Google Scholar
  6. G. Beer: Conjugate convex functions and the epi-distance topology, Proc. Amer. Math. Soc. 108 (1990) 117–126.https://doi.org/10.1090/S0002-9939-1990-0982400-8Google Scholar
  7. G. Beer: Topologies on Closed and Closed Convex Sets, Kluwer, Dordrecht (1993).https://doi.org/10.1007/978-94-015-8149-3Google Scholar
  8. G. Beer: On metric boundedness structures, Set-Valued Anal. 7 (1999) 195–208.https://doi.org/10.1023/A:1008720619545Google Scholar
  9. G. Beer, A. Di Concilio: Uniform continuity on bounded sets and the Attouch-Wets topology, Proc. Amer. Math. Soc. 112 (1991) 235–243.https://doi.org/10.1090/S0002-9939-1991-1033956-1Google Scholar
  10. G. Beer, S. Naimpally, J. Rodriguez-Lopez: S-topologies and bounded convergences, J. Math. Anal. Appl. 339 (2008) 542–552.https://doi.org/10.1016/j.jmaa.2007.07.010Google Scholar
  11. A. Caterino, T. Panduri, M. Vipera: Boundedness, one-point extensions, and B-extensions, Math. Slovaca 58(1) (2008) 101–114.https://doi.org/10.2478/s12175-007-0058-8Google Scholar
  12. S. Costantini, S. Levi, J. Zieminska: Metrics that generate the same hyperspace convergence, Set-Valued Anal. 1 (1993) 141–157.https://doi.org/10.1007/BF01027689Google Scholar
  13. L. Contesse, J.-P. Penot: Continuity of polarity and conjugacy for the epi-distance topology, J. Math. Anal. Appl. 156 (1991) 305–328.https://doi.org/10.1016/0022-247X(91)90399-KGoogle Scholar
  14. G. Di Maio, E. Meccariello, S. Naimpally: Uniformizing (proximal) △-topologies, Topology Appl. 137 (2004) 99–113.https://doi.org/10.1016/S0166-8641(03)00202-5Google Scholar
  15. R. Engelking: General Topology, Polish Scientific Publishers, Warsaw (1977).Google Scholar
  16. H. Hogbe-Nlend: Bornologies and Functional Analysis, North-Holland, Amsterdam (1977).Google Scholar
  17. S.-T. Hu: Boundedness in a topological space, J. Math. Pures Appl. 228 (1949) 287–320.Google Scholar
  18. J.-L. Joly: Une famille de topologies sur l’ensemble des fonctions convexes pour lesquelles la polarité est bicontinue, J. Math. Pures Appl. 52 (1973) 421–441.Google Scholar
  19. E. Klein, A. Thompson: Theory of Correspondences, Wiley, New York (1984).Google Scholar
  20. A. Lechicki, S. Levi, A. Spakowski: Bornological convergences, J. Math. Anal. Appl. 297 (2004) 751–770.https://doi.org/10.1016/j.jmaa.2004.04.046Google Scholar
  21. R. Lucchetti: Convexity and Well-Posed Problems, Springer, Berlin (2006).https://doi.org/10.1007/0-387-31082-7Google Scholar
  22. U. Mosco: On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971) 518–535.https://doi.org/10.1016/0022-247X(71)90200-9Google Scholar
  23. S. Naimpally, B. Warrack: Proximity Spaces, Cambridge Univ. Press, Cambridge (1970).https://doi.org/10.1017/CBO9780511569364Google Scholar
  24. J.-P. Penot: The cosmic Hausdorff topology, the bounded Hausdorff topology, and continuity of polarity, Proc. Amer. Math. Soc. 113 (1991) 275–286.https://doi.org/10.2307/2048468Google Scholar
  25. J.-P. Penot, C. Zalinescu: Bounded (Hausdorff) convergence: basic facts and applications, in: Variational Analysis and Applications, F. Giannessi, A. Maugeri (eds.), Springer, New York (2005).Google Scholar
  26. R. T. Rockafellar, R. Wets: Variational Analysis, 2nd Ed., Springer, New York (2004).Google Scholar
  27. A. Taylor, D. Lay: Introduction to Functional Analysis, Wiley, New York (1980).Google Scholar
  28. D. Walkup, R. Wets: Continuity of some convex-cone valued mappings, Proc. Amer. Math. Soc. 18 (1967) 229–235.https://doi.org/10.1090/S0002-9939-1967-0209806-6Google Scholar
  29. S. Willard: General Topology, Addison-Wesley, Reading (1970).Google Scholar

Suggested citation

G. Beer, S. Levi. “Pseudometrizable Bornological Convergence is Attouch-Wets Convergence.” Journal of Convex Analysis 15 (2008), No. 2, 439–453. https://doi.org/10.68381/jca15031

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2008. Rights now held by Banach Press.