Hyperspace Topologies

In this paper we study properties of metric spaces. We consider the collection of all nonempty closed subsets, Cl(X), of a metric space (X,d) and topologies on C.(X) induced by d. In particular, we investigate the Hausdorff topology and the Wijsman topology. Necessary and sufficient conditions are given for when a particular pseudo-metric is a metric in the Wijsman topology. The metric properties of the two topologies are compared and contrasted to show which also hold in the respective topologies. We then look at the metric space R-n, and build two residual sets. One residual set is the collection of uncountable, closed subsets of R-n and the other residual set is the collection of closed subsets of R-n having n-dimensional Lebesgue measure zero. We conclude with the intersection of these two sets being a residual set representing the collection of uncountable, closed subsets of R-n having n-dimensional Lebesgue measure zero.

Grantor dc:publisher

University of North Texas

Year dc:date

2001

Author dc:creator

• Freeman, Jeannette Broad

Contributors dc:contributor

• Bator, Elizabeth M.
• Lewis, Paul

• Metric spaces
• Topology
• Hausforff topology
• Wijsman topology
• Cl(X)
• properties

Statement dc:rights

• Public
• Copyright
• Freeman, Jeannette Broad
• Copyright is held by the author, unless otherwise noted. All rights reserved.

Language dc:language

English

Identifier

oclc: 51180548
https://digital.library.unt.edu/ark:/67531/metadc2902/
ark: ark:/67531/metadc2902

OAI identifier oai:identifier

info:ark/67531/metadc2902