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Topological Completeness of Function Spaces Arising in the Hausdorff Approximation of Functions

Published online by Cambridge University Press:  20 November 2018

Gerald Beer
Gerald Beer*
Affiliation:
Department of Mathematics California State University, Los Angeles Los Angeles, California 90032
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Abstract

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Let X be a complete metric space. Viewing continuous real functions on X as closed subsets of X × R, equipped with Hausdorff distance, we show that C(X, R) is completely metrizable provided X is complete and sigma compact. Following the Bulgarian school of constructive approximation theory, a bounded discontinuous function may be identified with its completed graph, the set of points between the upper and lower envelopes of the function. We show that the space of completed graphs, too, is completely metrizable, provided X is locally connected as well as sigma compact and complete. In the process, when X is a Polish space, we provide a simple answer to the following foundational question: which subsets of X × R arise as completed graphs?

Keywords

54C3554B2054E50function spaceHausdorff metriccompleted graphtopological completenesscomplete metric space
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 35 , Issue 4 , 01 December 1992 , pp. 439 - 448
Copyright
Copyright © Canadian Mathematical Society 1992

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