Skip to main content
SpringerLink
Log in

Uniform continuity of pointfree real functions via farness and related Galois connections

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

This paper concerns uniform continuity of real-valued functions on a (pre-)uniform frame. The aim of the paper is to characterize uniform continuity of such frame homomorphisms in terms of a farness relation between elements in the frame, and then to derive from it a separation and an extension theorem for real-valued uniform maps on uniform frames. The approach, purely order-theoretic, uses properties of the Galois maps associated with the farness relation. As a byproduct, we identify sufficient conditions under which a (continuous) scale in a frame with a preuniformity generates a real-valued uniform map.

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., Walters-Wayland, J.: \(C\)- and \(C^*\)-quotients in pointfree topology. Dissertationes Mathematicae (Rozprawy Mat.) 412, 1–62 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Banaschewski, B.: The real numbers in pointfree topology. In: Textos de Matemática, vol. 12. University of Coimbra, Coimbra (1997)

    Google Scholar 

  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 Acad. Publ., Dordrecht (2003)

    Google Scholar 

  4. Birkhoff, G.: Lattice Theory, 3rd edn. AMS Colloquium Publications (1967)

  5. Erné, M.: Adjunctions and Galois connections: origins, history and development. In: Denecke, K., Erné, M., Wismath, S.L. (eds.) Galois Connections and Applications, pp. 1–138. Kluwer Academic Publishers, Dordrecht–Boston–London (2004)

    MATH  Google Scholar 

  6. Gutiérrez García, J., Kubiak, T., Picado, J.: Perfect locales and localic real functions. Algebra Universalis 81, 32 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gutiérrez García, J., Picado, J.: On the parallel between normality and extremal disconnectedness. J. Pure Appl. Algebra 218, 784–803 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hušek, M.: Extension of mappings and pseudometrics. Extracta Math. 25, 277–308 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Kaiser, T.: A sufficient condition of full normality. Comment. Math. Univ. Carolin. 37, 381–389 (1996)

    MathSciNet  MATH  Google Scholar 

  10. Katětov, M.: On real valued functions in topological spaces. Fund. Math. 38, 85–91 (1951). (Fund. Math. 40, 139–142 (1953, Correction))

    Article  MathSciNet  MATH  Google Scholar 

  11. Kubiak, T.: On fuzzy topologies. Ph.D. Thesis, UAM, Poznań (1985)

  12. Kubiak, T.: Separation axioms: extension of mappings and embedding of spaces. In: Höhle, U., Rodabaugh, S.E. (eds.) Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, The Handbooks of Fuzzy Sets Series, vol. 3, pp. 433–479. Kluwer, Dordrecht (1999)

    Chapter  Google Scholar 

  13. Ore, O.: Galois connexions. Trans. Am. Math. Soc. 55, 493–513 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  14. Picado, J.: A new look at localic interpolation theorems. Topol. Appl. 153, 3203–3218 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Picado, J., Pultr, A.: Frames and locales: topology without points. In: Frontiers in Mathematics, vol. 28. Springer, Basel (2012)

    MATH  Google Scholar 

  16. Picado, J., Pultr, A.: Separation in Point-Free Topology. Birkhäuser/Springer, Cham (2021)

    Book  MATH  Google Scholar 

  17. Preiss, D., Vilimovský, J.: In-between theorems in uniform spaces. Trans. Am. Math. Soc. 261, 483–501 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pultr, A.: Pointless uniformities I. Comment. Math. Univ. Carolin. 25, 91–104 (1984)

    MathSciNet  MATH  Google Scholar 

  19. Smirnov, J.M.: On proximity spaces, (Russian). Mat. Sb. 31, 543–574 (1952). (English trans.: Amer. Math. Soc. Transl. (Ser. 2) 38, 5–35 (1964))

    Google Scholar 

  20. Walters-Wayland, J.: Completeness and nearly fine uniform frames. Ph.D. Thesis, University of Cape Town (1996)

Download references

Author information

Authors and Affiliations

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

    Ana Belén Avilez & Jorge Picado

Authors
  1. Ana Belén Avilez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jorge Picado
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jorge Picado.

Additional information

Communicated by Presented by W.Wm. McGovern.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors gratefully acknowledge partial financial support from the Centre for Mathematics of the University of Coimbra (UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES). The first author also acknowledges the support of a PhD grant from FCT/MCTES (PD/BD/150353/2019).

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Avilez, A.B., Picado, J. Uniform continuity of pointfree real functions via farness and related Galois connections. Algebra Univers. 83, 39 (2022). https://doi.org/10.1007/s00012-022-00795-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00012-022-00795-0

Keywords

Mathematics Subject Classification

Search

Navigation