Metric and metrizable mappings (Q388012)

The paper extends results about metric spaces to so-called metric mappings.NEWLINENEWLINEA simplified variant of the original definition of the concept of completeness of a metric mapping is used to construct its metric completion. The presented completion method is close to the standard method in metric spaces. Connections between completions, fibrewise completions and fibrewise complete extensions of metric mappings are explored. For closed metric mappings, all these extensions coincide. Furthermore the Lavrentieff theorem (about \(G_\delta\)-extensions of homeomorphisms between subsets of metric spaces) is extended to metric mappings. It is also proved that a uniformly continuous map-morphism of metric mappings can be extended to a uniformly continuous map-morphism of their completions. Finally, the Nagata-Smirnov and Bing metrization theorems are generalized to metric mappings.