Embedding metric families
J. F. McClendon · Pacific Journal of Mathematics · 1976
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.