Doitchinov, Doitchin A concept of completeness of quasi-uniform spaces. (English) Zbl 0723.54030 Topology Appl. 38, No. 3, 205-217 (1991). The author proposes a new definition of completeness for quasi-uniform spaces. He defines a quasi-uniform space (X,\({\mathcal U})\) to be quiet provided that it satisfies the following: For each \(U\in {\mathcal U}\) there is an entourage \(V\in {\mathcal U}\) such that if \(\{x'_{\alpha}|\alpha\in A\}\) and \(\{x''_{\beta},|\beta\in B\}\) are sets on X with \((x''_{\beta},x'_{\alpha})\to 0\) and \(x'\), \(x''\) are points of X such that \((x',x'_{\alpha})\in V\) for each \(\alpha\in A\) and \((x''_{\beta},x'')\in V\) for each \(\beta\in B\), then \((x',x'')\in U\). He shows that his theory of completeness is satisfactory for the class of quiet quasi-uniform spaces. Reviewer: W.F.Lindgren (Slippery Rock) Cited in 6 ReviewsCited in 26 Documents MSC: 54E15 Uniform structures and generalizations 54E52 Baire category, Baire spaces Keywords:Cauchy net; completion; completeness; quiet quasi-uniform spaces PDFBibTeX XMLCite \textit{D. Doitchinov}, Topology Appl. 38, No. 3, 205--217 (1991; Zbl 0723.54030) Full Text: DOI References: [1] Bourbaki, N., Topologie générale (1961), Hermann: Hermann Paris, Chapitres I et II · Zbl 0139.15904 [2] Császár, Á., Complete extensions of quasi-uniform spaces, (Novak, J., General Topology and its Relations to Modern Analysis and Algebra V. General Topology and its Relations to Modern Analysis and Algebra V, Proceedings Fifth Prague Topology Symposium, 1981 (1982), Heldermann: Heldermann Berlin), 104-113 · Zbl 0494.54022 [3] Doitchinov, D., On completeness in quasi-metric spaces, Topology Appl., 30, 127-148 (1988) · Zbl 0668.54019 [4] Doitchinov, D., On completeness of quasi-uniform spaces, C.R. Acad. Bulgare Sci., 41, 7, 5-8 (1988) · Zbl 0649.54015 [5] Fletcher, P.; Lindgren, W. F., Quasi-Uniform Spaces (1982), Marcel Dekker: Marcel Dekker New York · Zbl 0402.54024 [6] Keller, H. H., Die Limes-Uniformisierbarkeit der Limesräume, Math. Ann., 176, 334-341 (1968) · Zbl 0155.50302 [7] Sieber, J. L.; Pervin, W. J., Completeness in quasi-uniform spaces, Math. Ann., 158, 79-81 (1965) · Zbl 0134.41702 [8] Stoltenberg, R. A., A completion for a quasi-uniform space, Proc. Amer. Math. Soc., 18, 864-867 (1967) · Zbl 0158.41301 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.