Metrizable completions and covering properties (Q1100762)

The authors find, for some topological properties P, internal conditions on a metrizable space X guaranteeing that ``some (or every) metrizable completion of X has P''. For example: Some completion of X is (locally) compact iff X is (locally) separable and every completion of X is (locally) compact iff X is compact. The local compactness case of this ``every'' result was proved by the second-named author in [Am. Math. Monthly 94, 863-865 (1987). The main tool is Hausdorff's theorem on extending metrics from closed subspaces.