Skip to main content
SpringerLink
Log in

Entourages, Density, Cauchy Maps, and Completion

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

We study uniformities and quasi-uniformities (uniformities without the symmetry axiom) in the common language of entourages. The techniques developed allow for a general theory in which uniformities are the symmetric part. In particular, we have a natural notion of Cauchy map independent of symmetry and a very simple general completion procedure (perhaps more transparent and simpler than the usual symmetric one).

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. Ball, R.N., Pultr, A.: Extending semilattices to frames using sites and coverages. Math. Slovaca 64, 527–544 (2014)

    Article  MathSciNet  Google Scholar 

  2. Banaschewski, B.: Completion in pointfree topology. Lecture Notes in Mathematics. University of Cape Town (1996)

  3. Banaschewski, B.: Uniform completion in pointfree topology. In: Topological and Algebraic Structures in Fuzzy Sets. Trends Log. Stud. Log. Libr., vol. 20, pp. 19–56, Kluwer Academic Publishers, Boston (2003)

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Banaschewski, B., Pultr, A.: Completion and Samuel compactification of nearness and uniform frames. Port. Math. 69, 113–126 (2012)

    Article  MathSciNet  Google Scholar 

  9. Dowker, C.H., Strauss, D.: Sums in the category of frames. Houst. J. Math. 3, 7–15 (1977)

    MathSciNet  MATH  Google Scholar 

  10. Fletcher, P., Lindgren, W.F.: Quasi-Uniform Spaces. Marcel Dekker, New York (1982)

    MATH  Google Scholar 

  11. Frith, J., Hunsaker, W.: Completion of quasi-uniform frames. Appl. Categ. Struct. 7, 261–270 (1999)

    Article  MathSciNet  Google Scholar 

  12. Frith, J., Hunsaker, W., Walters-Wayland, J.: The Samuel compactification of a quasi-uniform frame. Topol. Proc. 23, 115–126 (1998)

    MathSciNet  MATH  Google Scholar 

  13. Hong, S.S., Kim, Y.K.: Cauchy completions of nearness frames. Appl. Categ. Struct. 3, 371–377 (1995)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Johnstone, P.T.: Stone Spaces. Cambridge Niversity Press, Cambridge (1982)

    MATH  Google Scholar 

  16. Kim, Y.K.: Cauchy completions of quasi-uniform frames. Kyungpook Math. J. 38, 429–437 (1998)

    MathSciNet  MATH  Google Scholar 

  17. Kříž, I.: A direct description of uniform completion in locales and a characterization of LT-groups Cahiers Topologie Géom. Différ. Catég. 27, 19–34 (1986)

    MATH  Google Scholar 

  18. Picado, J.: Weil uniformities for frames. Comment. Math. Univ. Carol. 36, 357–370 (1995)

    MathSciNet  MATH  Google Scholar 

  19. Picado, J.: Frame quasi-uniformities by entourages. In: Symposium on Categorical Topology, pp. 161–175. University of Cape Town, Rondebosch (1999)

  20. Picado, J.: Structured frames by Weil entourages. Appl. Categ. Struct. 8, 351–366 (2000)

    Article  MathSciNet  Google Scholar 

  21. Picado, J., Pultr, A.: Frames and Locales: Topology Without Points, Frontiers in Mathematics, vol. 28. Springer, Basel (2012)

    Book  Google Scholar 

  22. Picado, J., Pultr, A.: Entourages, covers and localic groups. Appl. Categ. Struct. 21, 49–66 (2013)

    Article  MathSciNet  Google Scholar 

  23. Picado, J., Pultr, A.: Notes on the product of locales. Math. Slovaca 65, 247–264 (2015)

    Article  MathSciNet  Google Scholar 

  24. Pultr, A., Tozzi, A.: Completion and coproducts of nearness frames. In: Symposium on Categorical Topology, pp. 177–186. University of Cape Town, Rondebosch (1999)

Download references

Acknowledgements

We are most grateful to the referee for all valuable comments and suggestions that improved the paper and, in particular, for alerting us for a mistake in the previous version of 6.2.1. Further, we acknowledge support from grants P202/12/G061 (Grant Agency of the Czech Republic) and MTM2015-63608-P (Ministry of Economy and Competitiveness of Spain) and from the Centre for Mathematics of the University of Coimbra (UID/MAT/00324/2013 funded by FCT/MCTES and FEDER through the Partnership Agreement PT2020).

Author information

Authors and Affiliations

  1. Department of Mathematics, CMUC, University of Coimbra, 3001-501, Coimbra, Portugal

    Jorge Picado

  2. Department of Applied Mathematics and ITI, MFF, Charles University, Malostranské nám. 24, 11800, Prague 1, Czech Republic

    Aleš Pultr

Authors
  1. Jorge Picado
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aleš Pultr
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jorge Picado.

Additional information

Communicated by B. Banaschewski.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Picado, J., Pultr, A. Entourages, Density, Cauchy Maps, and Completion. Appl Categor Struct 26, 1305–1324 (2018). https://doi.org/10.1007/s10485-018-9542-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-018-9542-2

Keywords

Mathematics Subject Classification

Search

Navigation