Techniques and examples in U-embedding
From abstract and introduction: ''A subset S of a metric space X is U- embedded in X if every uniformly continuous real-valued function on S extends to a uniformly continuous real-valued function on X. We consider the uniform extension question and examine the geometry of the problem. In Section 1 we give the definitions and notation which we will need. Section 2 gives some of the results which appear in the literature regarding extension of uniformly continuous functions. In Section 3 we present some general results on U-embedded sets. Section 4 deals with sets which determine boundedness, a concept which is useful in proving various sets are U-embedded. In Section 5 we characterize the U-embedded subsets of the real line. Section 6 gives a characterization of when the union of finitely many lines is U-embedded in \(R^ n\). Section 7 gives a large number of examples of sets in \(R^ n\) and \(R^ 3\) which are designed to indicate the subtle nature of the U-embedding property. Finally, Section 8 deals with U-embedding of the union of finitely many lines in the Lobachevskian plane.''
extension of uniformly continuous functions
scientific article; zbMATH DE number 3973799