The embedding of a metric
space in a Banach space plays an important role in metric space theory. In the
present paper we consider the problem of embedding a metric family X → D in a
Banach family. We obtain results under various hypotheses: (1) X a metric fiber
bundle, (2) X an extended metric family, and (3) X has the coarse topology for a
family of local cross-sections.
In §1 the basic definitions are given and a result is proved for metric fiber
bundles. In §2 some general conditions are given which suffice for embedding. §3
studies family metrics which are restrictions of continuous pseudometrics. §4
describes the topology of a metric family in terms of a given family of local sections.
In §5 a Banach family is associated with a given map and in §6 this is used to embed
a locally sectioned family. In §7 an example is described relating to the question of
embedding in a product family and also applying the techniques of §6 in a different
way.
