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

Beer, Gerald

Canadian mathematical bulletin, Tome 35 (1992) no. 4, pp. 439-448

Résumé

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?

Détail

DOI : 10.4153/CMB-1992-058-1

Mots-clés : 54C35, 54B20, 54E50, function space, Hausdorff metric, completed graph, topological completeness, complete metric space

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