A Strong Completeness Condition in Uniform Spaces with Well Ordered Bases

A natural strengthening of the completeness hypothesis, called B-completeness, allows us to extend to ωμ-metric spaces some properties of complete metric spaces which fail to hold if only the completeness condition is assumed. In this way we achieve some results on hyperspace theory, dealing with supercompleteness and with selections of multivalued functions. The notion of B-completeness is investigated and it is proved that quite important classes of ωμ-metric spaces have this property.