Skip to main content
SpringerLink
Log in

Quasi-Metrizable Spaces Satisfying Certain Completeness Conditions

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

It is shown that a quasi-metrizable space possesses a \(\lambda \)-base if and only if it admits a left \(K\)-complete quasi-metric and that a quasi-metrizable Moore space possesses a \(\lambda \)-base if and only if it admits a Smyth complete quasi-metric. Furthermore those quasi-metrizable Moore spaces are characterized that admit a right \(K\)-complete (equivalently, bicomplete) quasi-metric. Finally, it is established that a topological space admits a Smyth complete quasi-uniformity if and only if it is quasi-sober.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. Alemany and S. Romaguera, On right K-sequentially complete quasi-metric spaces, Acta Math. Hungar., 75 (1997), 267–278.

    Google Scholar 

  2. D. K. Burke, Covering properties, in: Handbook of Set-theoretic Topology, eds. K. Kunen and J. E. Vaughan, North-Holland (Amsterdam 1984), pp. 347–422.

    Google Scholar 

  3. Á. Császár, Regular extensions of quasi-uniformities, Studia Sci. Math. Hungar., 14 (1979), 15–26.

    Google Scholar 

  4. J. Deák and S. Romaguera, Co-stable quasi-uniform spaces, Ann. Univ. Sci. Budapest., 38 (1995), 55–70.

    Google Scholar 

  5. D. Doitchinov, On completeness in quasi-metric spaces, Topology Appl., 30 (1988), 127–148.

    Google Scholar 

  6. D. Doitchinov, A concept of completeness of quasi-uniform spaces, Topology Appl., 38 (1991), 205–217.

    Google Scholar 

  7. P. Fletcher, J. Hejcman and W. N. Hunsaker, A noncompletely regular quiet quasi-uniformity, Proc. Am. Math. Soc., 108 (1990), 1077–1079.

    Google Scholar 

  8. P. Fletcher and W. N. Hunsaker, Uniformly regular quasi-uniformities, Topology Appl., 37 (1990), 285–291.

    Google Scholar 

  9. P. Fletcher and W. N. Hunsaker, Completeness using pairs of filters, Topology Appl., 44 (1992), 149–155.

    Google Scholar 

  10. P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces, Lecture Notes Pure Appl. Math. 77, Dekker (New York, 1982).

    Google Scholar 

  11. L. Gillman and M. Jerison, Rings of Continuous Functions, Springer (New York, 1976).

    Google Scholar 

  12. H. J. K. Junnila, Neighbornets, Pacific J. Math., 76 (1978), 83–108.

    Google Scholar 

  13. J. Kofner, Transitivity and ortho-bases, Canad. J. Math., 33 (1981), 1439–1447.

    Google Scholar 

  14. H.-P. A. Künzi, A regular space without a uniformly regular quasi-uniformity, Monatshefte Math., 110 (1990), 115–116.

    Google Scholar 

  15. H.-P. A. Künzi, Nonsymmetric topology, in: Bolyai Society Math. Studies, 4, Topology with Applications (Szekszárd, Hungary, 1993), pp. 303–338.

  16. H.-P. A. Künzi, Preservation of completeness under mappings in asymmetric topology, Appl. General Topology, 1 (2000), 99–114.

    Google Scholar 

  17. H.-P. A. Künzi and G. C. L. Brümmer, Sobrification and bicompletion of totally bounded quasi-uniform spaces, Math. Proc. Camb. Phil. Soc., 101 (1987), 237–247.

    Google Scholar 

  18. H.-P. A. Künzi and N. Ferrario, Bicompleteness of the fine quasi-uniformity, Math. Proc. Camb. Phil. Soc., 109 (1991), 167–186.

    Google Scholar 

  19. H.-P. A. Künzi and H. J. K. Junnila, Stability in quasi-uniform spaces and the inverse problem, Topology Appl., 49 (1993), 175–189.

    Google Scholar 

  20. H.-P. A. Künzi and S. Romaguera, Some remarks on Doitchinov completeness, Topology Appl., 74 (1996), 61–72.

    Google Scholar 

  21. H.-P. A. Künzi and C. Ryser, The Bourbaki quasi-uniformity, Topology Proc., 20 (1995), 161–183.

    Google Scholar 

  22. M. J. Pérez-Peñalver and S. Romaguera, On right K-completeness of the fine quasi-uniformity, Questions Answers Gen. Topology, 14 (1996), 245–249.

    Google Scholar 

  23. S. Romaguera, Left K-completeness in quasi-metric spaces, Math. Nachr., 157 (1992), 15–23.

    Google Scholar 

  24. S. Romaguera, On complete Aronszajn quasi-metric spaces and subcompactness, in: Papers on General Topology and Applications, Eighth Summer Conf. at Queens College, 1992, Ann. New York Acad. Sci., 728 (1994), 114–121.

  25. S. Romaguera, On hereditary precompactness and completeness in quasi-uniform spaces, Acta Math. Hungar., 73 (1996), 159–178.

    Google Scholar 

  26. H. H. Wicke and J. M. Worrell, Jr., Open continuous mappings of spaces having bases of countable order, Duke Math. J., 34 (1967), 255–271; errata 813–814.

    Google Scholar 

  27. H. H. Wicke and J. M. Worrell, Jr., Topological completeness of first countable Hausdorff spaces I, Fund. Math., 75 (1972), 209–222.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS, UNIVERSITY OF CAPE TOWN, RONDEBOSCH, 7701, SOUTH AFRICA E-mail

    H.-P. A. Künzi

Authors
  1. H.-P. A. Künzi
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Künzi, HP.A. Quasi-Metrizable Spaces Satisfying Certain Completeness Conditions. Acta Mathematica Hungarica 95, 345–357 (2002). https://doi.org/10.1023/A:1015657410428

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1015657410428

Search

Navigation