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.
