A strong completeness condition in uniform spaces with well ordered bases (Q2716464)

The paper is concerned with the so-called \(\omega_\mu\)-metric spaces which may be considered as uniform spaces with well-ordered (by star-refinement) bases of coverings. These uniform spaces are natural generalizations of metric spaces but some properties of complete metric spaces cannot be extended to complete, in usual sense, \(\omega_\mu\)-metric spaces. For example, the generalizations of Michael's classical theorem on selection and the theorem on completeness of Hausdorff uniformity on the hyperspace of closed sets of complete metric space prove to be false. The authors prove in the paper that a strong completeness condition makes some generalizations of these theorems true.