Classical set convergences and topologies. (English) Zbl 0832.54012
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”.
A more detailed version of this paper is forthcoming under the title “Classical hyperspace topologies and convergences”.
Reviewer: M.Michael (Wilkes-Barre)
MSC:
| 54B20 | Hyperspaces in general topology |
| 54A10 | Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) |
Keywords:
Hausdorff topology; Attouch-Wets topology; Fell topology; Vietoris topology; Wijsman topology; Mosco topology; Kuratowski convergence; Kuratowski-Mosco convergenceReferences:
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