An embedding theorem for the Fell topology
type: Item
instance of: scholarly article
title: An embedding theorem for the Fell topology
zbMATH Open document ID: 0659.54008
published in: Michigan Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: Let X be a metric space, and let CL(X) be the set of all nonempty closed subsets of X. It is known that, when the metric d on X is bounded, \(E\mapsto d(\cdot,E)\) defines an embedding of CL(X) with the Hausdorff metric topology into the space of bounded continuous real-valued functions on X with the topology of uniform convergence. (In fact, it is an isometry with respect to the Hausdorff metric induced by d and the uniform metric.) This paper determines when the same formula gives an embedding of CL(X) with the weaker Fell topology [defined in the preceding review] into the space of continuous real-valued functions on X with the topology of uniform convergence on compacta. What is needed is a metric with respect to which X has nice closed balls (i.e., each noncompact closed ball is all of X), and such a metric is admissible iff X is locally compact. This result provides an alternate route to known facts about the Fell topology.
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54C25
zbMATH DE Number: 4078217
zbMATH Keywords: hyperspace of closed subsets
zbMATH Keywords: compact-open topology
zbMATH Keywords: Hausdorff metric topology
zbMATH Keywords: Fell topology
zbMATH Keywords: topology of uniform convergence on compacta
author: Gerald Beer
reviewed by: Mark Michael
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029003677
OpenAlex ID: W2010114501
DOI: 10.1307/MMJ/1029003677
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description: scientific article; zbMATH DE number 4078217