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On hereditary precompactness and completeness in quasi-uniform spaces. (English) Zbl 0924.54035

The author studies left (and right) \(K\)-complete quasi-uniform spaces. A quasi-uniformity on a set \(X\) is a filter \({\mathcal U}\) on \(X\times X\) such that for each \(U\in{\mathcal U}\), (i) \(\{(x,x):x\in X\}\subseteq U\) and (ii) \(V^2\subseteq U\) for some \(V\in{\mathcal U}\). A quasi-uniformity \({\mathcal U}\) on a set \(X\) determines the topology \(T({\mathcal U})\) on \(X\) in the standard way. Let \(X=(X,{\mathcal U})\) be a quasi-uniform space. Then, \(X\) is called uniformly regular if for each \(U\in{\mathcal U}\) there is \(V\in{\mathcal U}\) such that \(\operatorname{cl}_{T({\mathcal U})}V(x)\subseteq U(x)\) for each \(x\in X\). A filter \({\mathcal F}\) on \(X\) is called left (resp. right) \(K\)-Cauchy if for each \(U\in{\mathcal U}\) there is \(F\in{\mathcal F}\) such that \(U(x)\in{\mathcal F}\) (resp. \(U^{-1}(x)\in{\mathcal F}\)) for all \(x\in F\). Further, \(X\) is called left (resp. right) \(K\)-complete if every left (resp. right) \(K\)-Cauchy filter converges in the topology \(T({\mathcal U})\). Some of the main results are:
(1) A \(T_0\) uniformly regular quasi-uniform space is left \(K\)-complete if and only if it is Smyth-complete in the sense of M. B. Smyth [J. Lond. Math. Soc., II. Ser. 49, No. 2, 385-400 (1994; Zbl 0798.54036)];
(2) if a uniformly regular quasi-uniform space \((X,{\mathcal U})\) is left \(K\)-complete, then \((X,{\mathcal U}^{-1})\) is right \(K\)-complete, where \({\mathcal U}^{-1}=\{U^{-1}:U\in{\mathcal U}\}\).

MSC:

54E15 Uniform structures and generalizations

Citations:

Zbl 0798.54036
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[1] ?. Cs?sz?r, Extensions of quasi-uniformities, Acta Math. Hungar., 37 (1981), 121-145. · Zbl 0457.54020
[2] J. De?k, A biotopological view of quasi-uniform completeness I, II, III, Studia Sci. Math. Hungar., 30 (1995), 389-409; 30 (1995), 411-431; Part III to appear.
[3] J. De?k, Extending and completing quiet quasi-uniformities, Studia Sci. Math. Hungar., 29 (1994), 349-362. · Zbl 0848.54019
[4] J. De?k, Quasi-uniform completeness and neighbourhood filters, Studia Sci. Math. Hungar. (to appear). · Zbl 0859.54018
[5] J. De?k, On the coincidence of some notions of quasi-uniform completeness defined by filters pairs, Studia Sci. Math. Hungar., 26 (1991), 411-413. · Zbl 0778.54015
[6] J. De?k, A non-completely regular quiet quasi-metric, Math. Pannonica, 1 (1990), 111-116. · Zbl 0738.54009
[7] J. De?k, Letter to the author. June 1992.
[8] J. De?k and S. Romaguera, Co-stable quasi-uniform spaces, Ann. Univ. Sci. Budapest. (to appear). · Zbl 0856.54030
[9] D. Doitchinov, On completeness in quasi-metric spaces Top. Appl., 30 (1988), 127-148. · Zbl 0668.54019
[10] D. Doitchinov, On completeness of quasi-uniform spaces, C. R. Acad. Bulg. Sci., 41 (1988), 5-8. · Zbl 0668.54019
[11] D. Doitchinov, A concept of completeness of quasi-uniform spaces, Top. Appl., 38 (1991), 205-217. · Zbl 0723.54030
[12] D. Doitchinov, Another class of completable quasi-uniform spaces, C. R. Acad. Bulg. Sci., 44 (1991), 5-6. · Zbl 0776.54020
[13] P. Fletcher and W. Hunsaker, Uniformly regular quasi-uniformities, Top. Appl., 37, (1990), 285-291. · Zbl 0707.54021
[14] P. Fletcher and W. Hunsaker, Completeness using pairs of filters, Top. Appl., 44 (1992), 149-155. · Zbl 0770.54027
[15] P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces, Lecture Notes Pure Appl. Math. 77, Marcel Dekker, (New York, 1982). · Zbl 0501.54018
[16] J. R. Isbell, Uniform Spaces (Providence, 1964).
[17] R. D. Kopperman, Total boundedness and compactness for filter pairs, Ann. Univ. Sci. Budapest., 33 (1990), 25-30. · Zbl 0767.54021
[18] H. P. A. K?nzi, Nonsymmetric topology, in Proc. of Colloquium on Topology, 1993, Szeksz?rd, Hungary, Colloq. Math. Soc. J?nos Bolyai Math. Studies, 4 (1995), 303-338.
[19] H. P. A. K?nzi and H. J. K. Junnila, Stability in quasi-uniform spaces and the inverse problem, Top. Appl., 49 (1993), 175-189. · Zbl 0789.54034
[20] H. P. A. K?nzi, M. Mr?evi’c, I.L. Reilly and M. K. Vamanamurthy, Convergence, precompactness and symmetry in quasi-uniform spaces, Math. Japonica, 38 (1993), 239-253.
[21] I. L. Reilly, P. V. Subrahmanyam and M. K. Vamanamurthy, Cauchy sequences in quasi-pseudo-metric spaces, Monatsh. Math., 93 (1982), 127-140. · Zbl 0472.54018
[22] S. Romaguera, Left K-completeness in quasi-metric spaces, Math. Nachr., 157 (1992), 15-23. · Zbl 0784.54027
[23] S. Romaguera, Left K-complete quasi-uniform spaces, Seminarberichte Fachbereich Math. Fernuniversit?t Hagen, 48 (1994), 101-114.
[24] E. Alemany and S. Romaguera, On right K-sequentially complete quasi-metric spaces, Acta Math. Hungar. (to appear). · Zbl 0924.54037
[25] S. Romaguera and A. Guti?rrez, A note on Cauchy sequences in quasi-pseudometric spaces, Glasnik Mat., 21 (1986), 191-200.
[26] S. Salbany and S. Romaguera, On countably compact quasi-pseudo-metrizable spaces, J. Austral. Math. Soc. (Series A), 49 (1990), 231-240. · Zbl 0706.54027
[27] J. L. Sieber and W. J. Pervin, Completeness in quasi-uniform spaces, Math. Ann., 158, (1965), 79-81. · Zbl 0134.41702
[28] M. B. Smyth, Totally bounded spaces and compact ordered spaces as domains of computation, in Topology and Category Theory in Computer Science; ed. G. M. Reed, A. W. Roscoe and R. F. Wachter. Clarendon Press, (Oxford, 1991), 207-229.
[29] M. B. Smyth, Completeness of quasi-uniform and syntopological spaces, J. London Math. Soc., 49 (1994), 385-400. · Zbl 0798.54036
[30] R. A. Stoltenberg, Some properties of quasi-uniform spaces, Proc. London Math. Soc., 17 (1967), 226-240. · Zbl 0152.21003
[31] Ph. S?nderhauf, The Smyth completion of a quasi-uniform space, in Semantics of Programming Languages and Model Theory, ?Algebra, Logic and Applications? M. Droste and Y. Gurevich, eds., Gordon and Breach Sci. Publ. (New York, 1993), 189-212.
[32] Ph. S?nderhauf, Quasi-uniform completeness in terms of Cauchy nets, Acta Math. Hungar., 69 (1995), 47-54. · Zbl 0845.54016
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