On Complete Metrizability of Hyperspaces

UNCP Author/Contributor (non-UNCP co-authors, if there are any, appear on document)
Dr. Laszlo Zsilinszky, Professor (Creator)
The University of North Carolina at Pembroke (UNCP )
Web Site: http://www.uncp.edu/academics/library

Abstract: The purpose of the talk is to present new completeness results about the topology of bornological convergence S on the hyperspace C (X) of all nonempty closed subsets of a metric space (X; d). Two of the oldest and best applied hypertopologies, the Hausdor metric topology Hd and the Attouch-Wets topology AWd are prototypes of S. It is well-known that if (X; d) is a complete metric space, then both (CL(X); Hd ) and (CL(X); AWd ) are completely metrizable, however, since these topologies are sensitive to the metric of the base space X, complete metrizability of the hyperspaces may not automatically follow, if X is completely metrizable. Indeed, it will be demonstrated that a bounded completely metrizable space (X;d) exists, so that (CL(X); Hd ) contains a closed copy of the rationals. The methods used, involving topological games, then can be generalized to obtain a characterization of complete metrizablity of all topologies of bornological convergence. Other completeness properties of these hyperspaces, as well as some open problems, will be also discussed.

Additional Information

IX Iberoamerican Conference on Topology and Its Applications; Almería, Spain
Language: English
Date: 2014
Mathematics, Metrizability, Hyperspaces, Bornological Convergence, Topology, Hausdor Metric Topology, Attouch-Wets Topology, Rationals, Topological Games

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