Vol. 57, No. 2, 1975

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Metric families

J. F. McClendon

Vol. 57 (1975), No. 2, 491–509

A (continuous) metric family is a disjoint collection of metric spaces whose metrics are compatible with a given topology on the disjoint union. The purpose of this paper is to give some examples of these objects and to develop some of their basic properties. Most theorems about metric spaces can at least be formulated for metric families—some are true, some are true only with extra hypotheses, and some are false. Examples of each kind will be given. The main positive results are a version of Dugundji’s extension theorems, a cross section theorem, a generalization of one of Michael’s selection theorems, and a generalization of one of Coban’s selection theorems.

Mathematical Subject Classification 2000
Primary: 55G40
Secondary: 54C65
Received: 8 February 1974
Revised: 3 February 1975
Published: 1 April 1975
J. F. McClendon