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On convergence of closed sets in a metric space and distance functions

Published online by Cambridge University Press:  17 April 2009

Gerald Beer
Gerald Beer
Affiliation:
Department of Mathematics, California State University at Los Angeles, Los Angeles, California 90032, USA.
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Abstract

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Let CL(X) denote the nonempty closed subsets of a metric space X. We answer the following question: in which spaces X is the Kuratowski convergence of a sequence {Cn} in CL(X) to a nonempty closed set C equivalent to the pointwise convergence of the distance functions for the sets in the sequence to the distance function for C ? We also obtain some related results from two general convergence theorems for equicontinuous families of real valued functions regarding the convergence of graphs and epigraphs of functions in the family.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 31 , Issue 3 , June 1985 , pp. 421 - 432
Copyright
Copyright © Australian Mathematical Society 1985

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