A STRONG COMPLETENESS CONDITION IN UNIFORM SPACES WITH WELL ORDERED BASES
A natural strengthening of the completeness hypothesis, called -completeness, allows us to extend to ! -metric spaces some properties of complete metric spaces which fail to hold if only the completeness condi- tion is assumed. In this way we achieve some results on hyperspace theory, dealing with supercompleteness and with selections of multivalued functions. The notion of -completeness is investigated and it is proved that quite important classes of ! -metric spaces have this property.
Some results on ωμ-metric spaces
SummarySome concepts used in the valued spaces (spherical completeness and closure, pseudoconvergence) are introduced in the study of ωμ-metric spaces; in this context such notions are discussed and employed to show some extensions of theorems well- known in the metric spaces (contraction lemma, Baire's theorem, embedding of the generalized Cantor space); spherically closed sets are characterised by means of a retraction theorem.
Uniformities with totally ordered bases have paracompact topologies
This result does not appear to be in the literature, though the method of proof is related to those used in (1) and, particularly, (2) to prove A. H. Stone's theorem that metrics have paracompact topologies.