Morita-extensions and nearness-completions
The authors compare \textit{K. Morita's} notions of completeness and completion for \(T\)-uniform spaces [Proc. Japan Acad. 27, 65-72 (1951; Zbl 0042.41203), No. 3, 130-137, No. 4, 166-171 (1951; Zbl 0043.16504); 632-636 (1951; Zbl 0045.11702)] to more recent corresponding notions in the context of nearness spaces, which were introduced by the second author in [General Topol. Appl. 4, 191-212 (1974; Zbl 0288.54004)]. To facilitate the comparison, a nearness space is called Morita-complete if every strong Cauchy filter converges and is called complete if every cluster has an adherent point (equivalently, if every round Cauchy filter converges). Every complete nearness space is Morita-complete, and the notions coincide for regular spaces. However, there exist separated Morita-complete spaces which fail to be complete. The authors investigate some results concerning three classes of nearness spaces: regular (Morita-)complete spaces, separated complete spaces, and separated Morita-complete spaces. The first (respectively, second) class forms an epireflective subcategory of the category of regular (respectively, separated) nearness spaces, and its members are precisely those regular (respectively, separated) spaces which do not admit a proper dense regular (respectively, separated) extension. However, the third class does not even form a reflective subcategory of the category of nearness spaces, and there exist separated Morita-complete spaces which possess proper dense separated extensions. The authors also show that Morita's canonical completion of a nearness space can be regarded in a natural way as a subspace of the completion constructed by the second author in 1974. In the regular case these completions coincide, but they may fail to be equivalent in the separated case.
scientific article; zbMATH DE number 1121450
scientific article