On the completeness of non-symmetrical uniform convergence with some links to approach spaces
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The quasitopos b-UFIL of b-uniform filter spaces  are an appropri-ate tool for studying convergence from a higher point of view as demonstrated inrecent papers by the above mentioned authors. In addition BORN, the category ofbornological spaces and bounded maps, can be integrated as bicoreflective subcate-gory of b-UFIL. As already shown symmetric b-uniform filter spaces have “Cauchycompletions” which generalize some important ones as for example those which wereconsidered by Wyler, Preuss, Czászár and Hausdorff, respectively. In the presentpaper we will construct a completion, called ultracompletion, for a suitable notnecessarily symmetric b-uniform filter space and compare this one with a constructpresented for quasi-uniform spaces by Carlson and Hicks in the past. Furthermore,among others, we get the result that every quasiuniform limit space in the sense ofBehling has an ultracompletion. At the end of this article, we consider some impor-tant links to generalized approach spaces, those which were introduced by Lowen.So it is shown that b-topological closure operators can be completely described byso-called approach-bornologies, which represent a common generalization of bothapproach spaces and bornological spaces, respectively. Thus, as interesting corol-lary we obtain the result that APB the category of approach-bornological spacesand contracted maps intersects b-URING, the full subcategory of b-UFIL, whoseobjects have ultracompletions.
Document typePeer reviewed
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SourceMathematics for Applications. 2019 vol. 8, č. 1, s. 37-57. ISSN 1805-3629
- 2019/1