Abstract

This article proposes a new method for a unified description of hypertopologies on the closed subsets of an arbitrary topological space X. In the metric case, it represents a natural generalization of the description of hypertopologies as weak topologies induced by gap functionals. The tools are a family of sets of real valued functions on X and a family of subsets of X. This method allows us to recover most of the standard hypertopologies, and naturally leads us to the definition of new hypertopologies.

Author information

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

Roberto Lucchetti

Dipartimento di Matematica, Università di Milano, Via C. Saldini 50, I-20133 Milano, Italy.

Angela Pasquale

Department of Mathematics, University of Washington, Seattle, WA 98195, USA.

Keywords

  • hyperspace
  • gap functional
  • weak topologies
  • Fell topology
  • Wijsman topology
  • Vietoris topology
  • Hausdorff metric topology
  • Mosco topology

References

  1. G. Beer: Topologies on closed and closed convex sets, Kluwer Academic Publishers 1994, Dordrecht.https://doi.org/10.1007/978-94-015-8149-3Google Scholar
  2. G. Beer, R. Lucchetti: Weak topologies for the closed subsets of a metrizable space, Trans. Amer. Math. Soc. 335 (1993) 805–822.https://doi.org/10.1090/S0002-9947-1993-1094552-XGoogle Scholar
  3. G. Beer, R. Lucchetti: Convergence of epigraphs and sublevel sets, Set-Valued Analysis 1 (1993) 103–115.https://doi.org/10.1007/BF01027690Google Scholar
  4. G. Beer, A. Lechicki, S. Levi and S. Naimpally: Distance functionals and suprema of the hyperspace topologies, Annali Mat. Pura Appl. 162 (1992) 367–381.https://doi.org/10.1007/BF01760016Google Scholar
  5. J. Kelley: General Topology, Van Nostrand Company, Princeton, 1955.Google Scholar
  6. M.J. Kowalski: Topological spaces, New York Academic Press 1964.https://doi.org/10.1016/B978-1-4832-3316-1.50007-2Google Scholar
  7. R. Lucchetti, A. Pasquale: The bounded Vietoris topology and applications, Ricerche di Matematica XLIII, (1994) 93–110.Google Scholar
  8. R. Lucchetti, P. Shunmugaraj and Y. Sonntag: Recent hypertopologies and continuity of the value function and of the constrained level sets, Numer. Funct. Anal. and Optimiz. 14 (1993) 103–115.https://doi.org/10.1080/01630569308816510Google Scholar
  9. Y. Sonntag, C. Zǎlinescu: Set convergences. An attempt of classification., to appear, Trans. Amer. Math. Soc.Google Scholar
  10. Y. Sonntag, C. Zǎlinescu: Set convergence: a survey and a classification, to appear, Set-Valued Analysis.Google Scholar

Suggested citation

R. Lucchetti, A. Pasquale. “A New Approach to a Hyperspace Theory.” Journal of Convex Analysis 1 (1994), No. 2, 173–193. https://doi.org/10.68381/jca01013

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Published by Heldermann Verlag, 1994. Rights now held by Banach Press LLC.