Metric and metrizable mappings

A simplified variant of the definition of the completeness for metric mappings (that is closed to the standard definition of the completeness for metric spaces) is obtained. This result is used to construct the completions of metric mappings by a method close to the standard completion method for metric spaces. Relations between completions, fibrewise completions and fibrewise complete extensions of metric mappings are clarified. It is shown that for closed metric mappings all of these extensions coincide (since the completeness of these mappings coincides with their fibrewise completeness). The Lavrentieff theorem (about G δ -extensions of homeomorphisms between subsets of metric spaces) is extended to metric mappings. It is proved that a uniformly continuous map-morphism of metric mappings may be extended to a uniformly continuous map-morphism of their completions. The end of the paper contains generalizations of the Nagata–Smirnov and Bing metrization theorems for mappings.