Abstract
Two procedures for extending topological or uniform space concepts to bitopological or quasi-uniform spaces are: (1) spanning subcategories or functors by suitable objects; (2) lifting epireflections. The main theorem relates Cauchy completions of functorial admissible (quasi-) uniformities to generalized compactness reflections. We discuss the non-unique extension of the realcompactness reflection to bitopological spaces and the resulting bitopological version of Shirota's theorem.
AMS(MOS) codes
- Primary 54D60
- 54E15
- 54E55
- secondary 18A40
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Brümmer, G.C.L. Initial quasi-uniformities. Nederl. Akad. Wetensch. Proc. Ser. A 72=Indag.Math. 31 (1969), 403–409.
____ A categorial study of initiality in uniform topology. Thesis, Univ. Cape Town, 1971.
____ Struktuurfunktore en faktorisering. Proc.S.Afr. Math.Soc. 4 (1974), 81–83.
____ Topological functors and structure functors. Categorical Topology (Proc. Conf., Mannheim, 1975), pp.109–135. Lecture Notes in Math. 540, Springer-Verlag, Berlin, 1976.
____ On certain factorizations of functors into the category of quasi-uniform spaces. Quaestiones Math. 2(1977), 59–84.
____ On some bitopologically induced monads in Top. Mathematik-Arbeitspapiere Univ. Bremen, to appear.
____ and S. Salbany. On the notion of realcompactness for bitopological spaces. Math.Colloq. Univ.Cape Town 11(1977), 89–99.
Cooke, I.E. Epireflections in the category of bitopological spaces. Thesis, Univ. of London, 1972.
____ and I.L. Reilly. On bitopological compactness. J.London Math.Soc. (2) 9 (1975), 518–522.
Császár, Á. Fondements de la topologie générale. Akademiai Kiadó, Budapest, 1960. Revised and extended edition: Foundations of general topology. Pergamon Press, Oxford-New York, 1963.
____ Doppeltkompakte bitopologische Räume. In: G. Asser, J. Flachsmeyer and W. Rinow (ed.): Theory of Sets and Topology, pp.59–67. VEB Deutscher Verlag d.Wiss., Berlin, 1972.
Gillman, L. and M. Jerison. Rings of continuous functions. Van Nostrand, Princeton-New York, 1960.
Harvey, J.M. T0-separation in topological categories. Quaestiones Math. 2(1977), 177–190.
Hoffmann, R.-E. (E,M)-universally topological functors. Habilitationsschrift, Univ.Düsseldorf, 1974.
Kelly, J.C. Bitopological spaces. Proc.London Math.Soc. (3) 13 (1963),71–89.
Lane, E.P. Bitopological spaces and quasi-uniform spaces. Proc. London Math.Soc. (3)17(1967),241–256.
Murdeshwar, M.G. Bibliography on quasi-uniform spaces. Preprint, Dept.of Math., Univ. of Alberta, Edmonton, 1974.
Nachbin, L. Sur les espaces uniformes ordonnés. C.R. Acad.Sci. Paris 226(1948), 774–775.
Saegrove, M.J. Pairwise complete regularity and compactification in bitopological spaces. J.London Math.Soc. (2)7(1973),286–290.
Salbany, S. Quasi-uniformities and quasi-pseudometrics. Math. Colloq. Univ. Cape Town 6(1970–71), 88–102.
____ Bitopological spaces, compactifications and completions. Thesis, Univ. Cape Town, 1970. Reprinted as Math. Monogr. Univ. Cape Town No. 1, 1974.
____ Compactifications of bitopological spaces. Math. Colloq. Univ. Cape Town 7(1971–72), 1–3.
____ On quasi-uniformizability. Joint Math. Colloq. Univ. of South Africa and Univ. of the Witwatersrand 1973–74.
____ On compact bitopological spaces. Proc.S.Afr.Math.Soc. 4(1974), 219–223.
____ Completions and triples. Math.Colloq. Univ. Cape Town 8(1973), 55–61.
____ Reflective subcategories and closures operators. Categorical Topology (Proc.Conf., Mannheim, 1975), pp.549–565. Lecture Notes in Math. 540, Springer-Verlag, Berlin, 1976.
Shirota, T. A class of topological spaces. Osaka Math. J. 4 (1952), 23–40.
Swart, J. Total disconnectedness in bitopological spaces and product bitopological spaces. Nederl.Akad.Wetensch., Proc. Ser.A 74=Indag Math. 33(1971), 135–145.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1979 Springer-Verlag
About this paper
Cite this paper
Brümmer, G.C.L. (1979). Two procedures in bitopology. In: Herrlich, H., Preuß, G. (eds) Categorical Topology. Lecture Notes in Mathematics, vol 719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0065256
Download citation
DOI: https://doi.org/10.1007/BFb0065256
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-09503-3
Online ISBN: 978-3-540-35193-1
eBook Packages: Springer Book Archive
