Abstract
We characterize quasi-metrizable bispaces that admit only bicomplete quasimetrics by means of doubly primitive sequences, and deduce that if (X, S, T) is a quasi-metrizable bispace admitting only bicomplete quasi-metrics and either (X, S) or (X, T) is hereditarily Lindelöf, then (X, S ∨ T) is compact. We also give an example which shows that hereditary Lindelöfness cannot be omitted in the above result. Finally, we show that a quasi-pseudometrizable bispace (X, S, T) admits only totally bounded quasi-pseudometrics if and only if (X, S ∨ T) is compact, and deduce that a quasi-pseudometrizable topological space admits only totally bounded quasi-pseudometrics if and only if it is hereditarily compact and quasi-sober (equivalently, if and only if it admits a unique quasi-uniformity).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bourbaki N.,Algèbre Commutative, Hermann, (1961).
Brümmer G. C. L.,Initial quasi-uniformities, Indag. Math.,31 (1969), 403–409.
Fell J. M. G.,A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc.,13 (1962), 472–476.
Fletcher P.—Lindgren W. F.,Quasi-Uniform Spaces, Marcel Dekker, (1982).
Hausdorff F.,Erweiterung einer Homöomorphie, Fund. Math.,16 (1930), 353–360.
Hoffmann R.-E.,On the sobrification remainder s X/X, Pacific J. Math.,83 (1979), 145–156.
Junnila H. J. K.,Covering Properties and Quasi-Uniformities of Topological Spaces, Ph. D. Thesis, Blacksburg, Virginia, (1978).
Kelly J. C.,Bitopological spaces, Proc. London Math. Soc.,13 (1963), 71–89.
Künzi H. P. A.,On strongly quasi-metrizable spaces, Arch. Math. (Basel),41 (1983), 57–63.
Künzi H. P. A.,Functorial admissible quasi-uniformities on topological spaces, Topology Appl.,43 (1992), 27–36.
Künzi H. P. A.—Brümmer G. C. L.,Sobrification and bicompletion of totally bounded quasi-uniform spaces, Math. Proc. Cambridge Phil. Soc.,101 (1987), 237–247.
Künzi H. P. A.—Ferrario N.,Bicompleteness of the fine quasi-uniformity, Math. Proc. Cambridge Phil. Soc.,109 (1991), 167–186.
Künzi H. P. A.—Mršević M.—Reilly I. L.—Vamanamurthy M. K.,Convergence, precompactness and symmetry in quasi-uniform spaces, Math. Japonica,38 (1993), 239–253.
Künzi H. P. A.—Romaguera S.—Salbany S.,Topological spaces that admit bicomplete quasi-pseudometrics, Ann. Univ. Sci. Budapest,37 (1994), 185–195.
Lindgren W. F.,Topological spaces with a unique compatible quasi-uniformity, Canad. Math. Bull.,14 (1971), 369–372.
Romaguera S.—Salbany S.,On bicomplete quasi-pseudometrizability, Topology Appl.,50 (1993), 283–289.
Romaguera S.—Salbany S.,Dieudonné complete bispaces, Studia Sci. Math. Hungar., to appear.
Salbany S.—Romaguera S.,On countably compact quasi-pseudometrizable spaces, J. Austral. Math. Soc. (Series A),49 (1990), 231–240.
Skula L.,On a reflective subcategory of the category of all topological spaces, Trans. Amer. Math. Soc.,142 (1969), 37–41.
Stone A. H.,Hereditarily compact spaces, Amer. J. Math.,82 (1960), 900–916.
Author information
Authors and Affiliations
Additional information
The first author acknowledges support from the Swiss national Science Foundation, under grant 2000-056811.99 and from the Universidad Politécnica de Valencia. The second author acknowledges the support of the DGES, under grant PB95-0737. The third author acknowledges the support of the Universidad Politécnica de Valencia, as well as the provision of leave by the University of South Africa.
Rights and permissions
About this article
Cite this article
Künzi, H.P.A., Romaguera, S. & Salbany, S. Bispaces admitting only bicomplete or only totally bounded quasi-metrics. Rend. Circ. Mat. Palermo 50, 186–198 (2001). https://doi.org/10.1007/BF02843928
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02843928