Uniform continuity in ωμ-metric spaces and uc ωμ-metric extendability

Generally, the term uc-ness means some continuity is uniform. A metric space X is uc when any continuous function fromX to [0, 1] is uniformly continuous and a metrizable space X is a Nagata space when it can be equipped with a uc metric. We consider natural forms of uc-ness for the -metric spaces, which fill a very large and interesting class of uniform spaces containing the usual metric ones, and extend to them various different formulations of the metric uc-ness, by additionaly proving their equivalence. Furthermore, since any -compact space is uc and any uc -metric space is complete, in the line of constructing dense extensions which preserve some structure, such as uniform completions, we focus on the existence for an -metrizable space of dense topological extensions carrying a uc -metric. In this paper we show that an -metrizable space X is uc-extendable if and only if there exists a compatible -metric d on X such that the set X' of all accumulation points in X is crowded, i.e., any -sequence in X' has a d-Cauchy -subsequence in X'.