Classical set convergences and topologies
type: Item
instance of: scholarly article
title: Classical set convergences and topologies
zbMATH Open document ID: 0832.54012
DOI: 10.1007/BF01027103
author: Roberto Lucchetti
author: Anna Torre
published in: Set-Valued Analysis
publication date: 1996-03-05T00:00:00Z
review text: A great number of different topologies have been defined for the hyperspace of (nonempty) closed subsets of a metric space. The topologies referred to in the title are those which, in the author's words, ``have been `on the market' for a long time,'' namely, the Hausdorff, Attouch- Wets, Vietoris, Fell, Wijsman, and Mosco topologies (the last one being for the hyperspace of closed convex subsets of a reflexive Banach space). Most of the results in the paper are also classical. The presentation of how these topologies compare or fail to compare is streamlined by an efficient use of upper and lower topologies. Also compared are the Kuratowski and Kuratowski-Mosco convergences. The latter convergence is shown not to be topological if the base space is infinite-dimensional (a new result). The paper also explores the (complete) metrizability and related properties of the various hyperspaces. A more detailed version of this paper is forthcoming under the title ``Classical hyperspace topologies and convergences''.
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54A10
zbMATH DE Number: 627473
zbMATH Keywords: Hausdorff topology
zbMATH Keywords: Attouch-Wets topology
zbMATH Keywords: Fell topology
zbMATH Keywords: Vietoris topology
zbMATH Keywords: Wijsman topology
zbMATH Keywords: Mosco topology
zbMATH Keywords: Kuratowski convergence
zbMATH Keywords: Kuratowski-Mosco convergence
reviewed by: Mark Michael
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01027103
OpenAlex ID: W1990735641
description: scientific article; zbMATH DE number 627473