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Mapping Ideals to Sublocales

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Abstract

A map that is well known in C(X), dating back to the work of L. Gillman, M. Henriksen and M. Jerison in 1954, is here used to construct a morphism of quantales, whose right adjoint we show to be the map that sends a subset A of \(\beta X\) to the ideal

$$\begin{aligned} {\varvec{O}}^A =\{f\in C(X)\mid A\subseteq {{\,\mathrm{int}\,}}_{\beta X}{{\,\mathrm{cl}\,}}_{\beta X}Z(f)\} \end{aligned}$$

of C(X) if and only if X is a P-space. We show that when viewed this way, this map characterizes R.G. Woods’ WN-maps in terms of commutativity of a certain diagram in the category of quantales. All this is achieved most economically by working with locales instead of topological spaces. The Lindelöf reflection allows us to present “countable” analogues of the results just mentioned and we highlight the similarities and disparities.

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References

  1. Atalla, R.E.: \(P\)-sets in \(F^\prime \)-spaces. Proc. Am. Math. Soc. 46, 125–132 (1974)

    MathSciNet  MATH  Google Scholar 

  2. Azarpanah, F.: Algebraic properties of some compact spaces. Real Anal. Exchange 25, 317–328 (2000)

    Article  MathSciNet  Google Scholar 

  3. Ball, R.N., Walters-Wayland, J.: \(C\)- and \(C^{*}\)-quotients in pointfree topology. Dissert. Math. (Rozprawy Mat.) 412, 62pp (2002)

  4. Banaschewski, B.: The real numbers in pointfree topology. Textos de Matemática Série B, No. 12, Departamento de Matemática da Universidade de Coimbra (1997)

  5. Banaschewski, B., Gilmour, C.: Pseudocompactness and the cozero part of a frame. Comment. Math. Univ. Carolin. 37, 579–589 (1996)

    MathSciNet  MATH  Google Scholar 

  6. Dietrich, W.E.: On the ideal structure of \(C(X)\). Trans. Amer. Math. Soc. 152, 61–77 (1970)

    MathSciNet  MATH  Google Scholar 

  7. Dube, T.: Concerning \(P\)-frames, essential \(P\)-frames, and strongly zero-dimensional frames. Algebra Universalis 61, 115–138 (2009)

    Article  MathSciNet  Google Scholar 

  8. Dube, T.: Remote points and the like in pointfree topology. Acta Math. Hungar. 123, 203–222 (2009)

    Article  MathSciNet  Google Scholar 

  9. Dube, T.: Some ring-theoretic properties of almost \(P\)-frames. Algebra Universalis 60, 145–162 (2009)

    Article  MathSciNet  Google Scholar 

  10. Dube, T.: On the ideal of functions with compact support in pointfree function rings. Acta Math. Hungar. 129, 205–226 (2010)

    Article  MathSciNet  Google Scholar 

  11. Dube, T.: Concerning \(P\)-sublocales and disconnectivity. Appl. Categ. Structures 27, 365–383 (2019)

    Article  MathSciNet  Google Scholar 

  12. Dube, T.: On the maximal regular ideal of pointfree function rings, and more. Topology Appl. 273, Article 106960 (2020)

  13. Gillman, L., Henriksen, M., Jerison, M.: On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions. Proc. Amer. Math. Soc. 5, 447–455 (1954)

    Article  MathSciNet  Google Scholar 

  14. Gillman, L., Jerison, M.: Rings of Continuous Functions. Van Nostrand, Princeton (1960)

    Book  Google Scholar 

  15. Johnson, D.G., Mandelker, M.: Functions with pseudocompact support. Gen. Topology Appl. 3, 331–338 (1971)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  17. Levy, R.: Almost-P-spaces. Can. J. Math. 29, 284–288 (1977)

    Article  Google Scholar 

  18. Madden, J., Vermeer, J.: Lindelöf locales and realcompactness. Math. Proc. Camb. Phil. Soc. 99, 473–480 (1986)

    Article  Google Scholar 

  19. Mulvey, C.J.: Suppl. Renc. Circ. Mat. Palermo Ser. II(12), 99–104 (1986)

    Google Scholar 

  20. Picado, J., Pultr, A.: Frames and Locales: Topology without Points. Frontiers in Mathematics. Springer, Basel (2012)

    Book  Google Scholar 

  21. Picado, J., Pultr, A., Tozzi, A.: Joins of closed sublocales. Houst. J. Math. 45, 21–38 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Rosenthal, K.I.: Quantales and their applications. pitman research notes in mathematics series, vol. 234. Wiley, New York (1990)

    Google Scholar 

  23. Woods, R.G.: Maps that characterize normality properties and pseudocompactness. J. London Math. Soc. 7, 453–461 (1973)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We are most grateful to the two referees for some questions and helpful suggestions that have improved the first version of this paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003, Pretoria, South Africa

    Themba Dube & Dorca Nyamusi Stephen

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  1. Themba Dube
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  2. Dorca Nyamusi Stephen
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Correspondence to Themba Dube.

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Communicated by Jorge Picado.

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Dube, T., Stephen, D.N. Mapping Ideals to Sublocales. Appl Categor Struct 29, 747–772 (2021). https://doi.org/10.1007/s10485-021-09634-0

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