Skip to main content
SpringerLink
Log in

Enough Regular Cauchy Filters for Asymmetric Uniform and Nearness Structures

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

Quasi-nearness biframes provide an asymmetric setting for the study of nearness; in Frith and Schauerte (Quaest Math 33:507–530, 2010) a completion (called a quasi-completion) was constructed for such structures and in Frith and Schauerte (Quaest Math, 2012) completeness was characterized in terms of the convergence of regular Cauchy bifilters. In this paper questions of functoriality for this quasi-completion are considered and one sees that having enough regular Cauchy bifilters plays an important rôle. The quasi-complete strong quasi-nearness biframes with enough regular Cauchy bifilters are seen to form a coreflective subcategory of the strong quasi-nearness biframes with enough regular Cauchy bifilters. Here a significant difference between the symmetric and asymmetric cases emerges: a strong (even quasi-uniform) quasi-nearness biframe need not have enough regular Cauchy bifilters. The Cauchy filter quotient leads to further characterizations of those quasi-nearness biframes having enough regular Cauchy bifilters. The fact that the Cauchy filter quotient of a totally bounded quasi-nearness biframe is compact shows that any totally bounded quasi-nearness biframe with enough regular Cauchy bifilters is in fact quasi-uniform. The paper concludes with various examples and counterexamples illustrating the similarities and differences between the symmetric and asymmetric cases.

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. Banaschewski, B.: Completion in pointfree topology. In: Lecture Notes in Math. and Applied Math., no. 2. University of Cape Town (1996)

  2. Banaschewski, B.: Uniform completion in pointfree topology. In: Rodabaugh, S.E., Klement, E.P. (eds.) Topological and Algebraic Structures in Fuzzy Sets, pp. 19–56. Kluwer Academic, Norwell (2003)

    Chapter  Google Scholar 

  3. Banaschewski, B., Brümmer, G.C.L.: Strong zero-dimensionality of biframes and bispaces. Quaest. Math. 13, 273–290 (1990)

    Article  MATH  Google Scholar 

  4. Banaschewski, B., Pultr, A.: Samuel compactification and completion of uniform frames. Math. Proc. Camb. Philos. Soc. 108, 63–78 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Banaschewski, B., Pultr, A.: Cauchy points of uniform and nearness frames. Quaest. Math. 19, 101–127 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Banaschewski, B., Brümmer, G.C.L., Hardie, K.A.: Biframes and bispaces. Quaest. Math. 6, 13–25 (1983)

    Article  MATH  Google Scholar 

  7. Banaschewski, B., Hong, S.S., Pultr, A.: On the completion of nearness frames. Quaest. Math. 21, 19–37 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Frith, J.L.: Structured frames. Ph.D. Thesis, University of Cape Town (1987)

  9. Frith, J., Schauerte, A.: The Samuel compactification for quasi-uniform biframes. Topol. its Appl. 156, 2116–2122 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Frith, J., Schauerte, A.: Quasi-nearnesses on biframes and their completions. Quaest. Math. 33, 507–530 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Frith, J., Schauerte, A.: An asymmetric characterization of the congruence frame. Topol. its Appl. 158(7), 939–944 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Frith, J., Schauerte, A.: The reals as full and balanced biframe. Topol. its Appl. 158(17), 2322–2331 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Frith, J., Schauerte, A.: An asymmetric approach to filters in strict extensions and quotients. Acta Math. Hung. (2012). doi:10.1007/s10474-012-0208-5

    MathSciNet  Google Scholar 

  14. Frith, J., Schauerte, A.: Quasi-nearness biframes: unique completions and related covering properties. Quaest. Math. 35(1), 35–56 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Frith, J., Schauerte, A.: Asymmetric filter convergence and completeness. Quaest. Math. (2012, to appear)

  16. Isbell, J.R.: Atomless parts of spaces. Math. Scand. 31, 5–32 (1972)

    MathSciNet  MATH  Google Scholar 

  17. Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)

    MATH  Google Scholar 

  18. Kříž, I.: A direct description of uniform completion and a characterization of LT groups. Cah. Top. Géom. Différ. Catég. 27, 19–34 (1986)

    MATH  Google Scholar 

  19. Picado, J., Pultr, A.: Cover quasi-uniformities in frames. Topol. its Appl. 158(7), 869–881 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Picado, J., Pultr, A.: On strong inclusions and asymmetric proximities in frames. Order (2011). doi:10.1007/s11083-011-9218-0

    Google Scholar 

  21. Picado, J., Pultr, A.: Frames and Locales. Springer, Basel (2012)

    Book  MATH  Google Scholar 

  22. Picado, J., Pultr, A., Tozzi, A.: Locales. In: Pedicchio, M.C., Tholen, W. (eds.) Categorical Foundations. Encyclopedia of Mathematics and its Applications 97, pp. 49–101. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  23. Pultr, A.: Frames. In: Hazewinkel, M. (ed.) Handbook of Algebra, vol. 3, pp. 791–857. North-Holland, Amsterdam (2003)

  24. Schauerte, A.: Biframe compactifications. Comment. Math. Univ. Carol. 34(3), 567–574 (1993)

    MathSciNet  MATH  Google Scholar 

  25. Schauerte, A.: Normality for biframes. Appl. Categ. Struct. 3, 1–9 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  26. Schauerte, A.: Notions of local compactness and smallest compactifications of biframes. Appl. Categ. Struct. 14, 259–272 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Vickers, S.: Topology via logic. In: Cambridge Tracts in Theoretical Computer Science 5. Cambridge University Press, Cambridge (1989)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, Cape Town, South Africa

    John Frith & Anneliese Schauerte

Authors
  1. John Frith
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anneliese Schauerte
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anneliese Schauerte.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Frith, J., Schauerte, A. Enough Regular Cauchy Filters for Asymmetric Uniform and Nearness Structures. Appl Categor Struct 21, 681–701 (2013). https://doi.org/10.1007/s10485-012-9286-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-012-9286-3

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation