On bicomplete quasi-pseudometrizability. (English) Zbl 0827.54020

Romaguera, Salvador; Salbany, Sergio

Topology Appl. 50, No. 3, 283-289 (1993).

Summary: We present a characterization of those quasi-pseudometrizable bitopological spaces which admit a compatible bicomplete quasi- pseudometric. From this result we deduce several generalizations of classical theorems of Alexandroff, Hausdorff, Čech and Frolík.

MSC:

54E55	Bitopologies
54E50	Complete metric spaces

Keywords:

absolute \(G_ \delta\); bicompactification; Čech complete; quasi- pseudometrizable bitopological spaces; bicomplete quasi-pseudometric

References:

[1]	Alexandroff, P. S., Sur les ensembles de la première classe et les espaces abstraits, C.R. Acad. Sci. Paris. Sér. I. Math., 178, 185-187 (1924) · JFM 50.0134.01
[2]	Archangel’skii, A., On topological spaces which are complete in the sense of Čech, Becth. Mock. Yh. Cep. Mat., 2, 37-40 (1961), (in Russian)
[3]	Čech, E., On biocompact spaces, Ann. of Math., 38, 823-844 (1937) · JFM 63.0570.02
[4]	Dieudonné, J., Sur un espace localement compact non metrisable, An. Acad. Brasil. Ciênc., 19, 67-69 (1947)
[5]	Engelking, R., General Topology, (Monografie Matematyczne, 60 (1977), PWN: PWN Warsaw) · Zbl 0684.54001
[6]	Fletcher, P.; Lindgren, W. F., Quasi-Uniform Spaces (1982), Marcel Dekker: Marcel Dekker New York · Zbl 0402.54024
[7]	Frolík, Z., Generalizations of the \(G_σ\)-property of complete metric spaces, Czechoslovak Math. J., 10, 359-379 (1960) · Zbl 0100.18701
[8]	Hausdorff, F., Die Mengen \(G_σ\) in vollständidgen Räumen, Fund. Math., 6, 146-148 (1924)
[9]	Kelley, J. L., General Topology (1955), Springer: Springer New York · Zbl 0066.16604
[10]	Kelly, J. C., Bitopological spaces, Proc. London Math. Soc., 13, 71-89 (1963) · Zbl 0107.16401
[11]	Lane, E. P., Bitopological spaces and quasi-uniform spaces, Proc. London Math. Soc., 17, 341-356 (1967) · Zbl 0152.21101
[12]	Romaguera, S.; Salbany, S., On the Niemytzki-Tychonoff theorem for distance spaces, Fasc. Math., 19, 233-241 (1990) · Zbl 0764.54021
[13]	Salbany, S., Bitopological spaces, compactifications and completions, (Math. Monographs, 1 (1974), University of Cape Town) · Zbl 0353.54021
[14]	Salbany, S., A bitopological view of topology and order, (Categorical Topology, Proceedings Toledo, OH, 1983 (1984), Heldermann: Heldermann Berlin), 481-504 · Zbl 0964.54033
[15]	Stoltenberg, R. A., On quasi-metric spaces, Duke Math. J., 36, 65-71 (1969) · Zbl 0176.51902