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• 作者：A. Di Concilio ; C. Guadagni
• 关键词：Key words and phrasesNagata extension ; uniform space ; uniform space with a totally ordered base ; \({\omega_\mu}\) ; metric space ; uc ; ness in \({\omega_\mu}\) ; metric space ; \({\omega_\mu}\) ; additive topological space ; Hausdorff–Bourbaki uniformity ; Hausdorff convergence ; Vietoris topology ; pseudo ; Cauchy net
• 刊名：Acta Mathematica Hungarica
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：150
• 期：1
• 页码：153-166
• 全文大小：774 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Sciences
  Mathematics
  
• 出版者：Akad茅miai Kiad贸, co-published with Springer Science+Business Media B.V., Formerly Kluwer Academic
• ISSN：1588-2632
• 卷排序：150

文摘

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 \({\omega_\mu}\)-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 \({\omega_\mu}\)-compact space is uc and any uc \({\omega_\mu}\)-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 \({\omega_\mu}\)-metrizable space of dense topological extensions carrying a uc \({\omega_\mu}\)-metric. In this paper we show that an \({\omega_\mu}\)-metrizable space X is uc-extendable if and only if there exists a compatible \({\omega_\mu}\)-metric d on X such that the set X′ of all accumulation points in X is crowded, i.e., any \({\omega_\mu}\)-sequence in X′ has a d-Cauchy \({\omega_\mu}\)-subsequence in X′.