Applied Categorical Structures

, Volume 27, Issue 6, pp 687–702 | Cite as

Maximal Lindelöf Locales

  • Themba DubeEmail author


For a subfit frame L, let \({\mathcal {S}}_{\mathfrak {c}}(L)\) denote the complete Boolean algebra whose elements are the sublocales of L that are joins of closed sublocales. Identifying every element of L with the open sublocale it determines allows us to view L as a subframe of \({\mathcal {S}}_{\mathfrak {c}}(L)\). With this backdrop, we say L is maximal Lindelöf if it is Lindelöf and whenever \(L\subseteq M\), for some Lindelöf subframe M of \({\mathcal {S}}_{\mathfrak {c}}(L)\), then \(L=M\). Recall that a topological space \((X,\tau )\) is maximal Lindelöf if it is Lindelöf, and there is no strictly finer topology \(\rho \) on X such that \((X,\rho )\) is a Lindelöf space. We show that a space is maximal Lindelöf if and only if the frame of its open subsets is maximal Lindelöf. We then characterize maximal Lindelöf frames internally. Among regular frames, we show that the maximal Lindelöf frames are precisely the Lindelöf ones in which every \(F_\sigma \)-sublocale (meaning a join of countably many closed sublocales) is closed.


Frame Subframe Locale Sublocale Maximal-Lindelöf frame Localic map Closed localic map Open localic map 

Mathematics Subject Classification

Primary 06D22 Secondary 54A10 54D20 


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The author is greatly indebted to the referee for pointing out some crucial omissions and also for several helpful comments that resulted in a much improved paper.


  1. 1.
    Ball, R.N., Picado, J., Pultr, A.: Notes on exact meets and joins. Appl. Categ. Struct. 22, 699–714 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ball, R.N., Walters-Wayland, J.: \(C\)- and \(C^{*}\)-quotients in pointfree topology. Dissertation Mathematics (Rozprawy Matematyczne), vol. 412, 62 pp (2002)Google Scholar
  3. 3.
    Banaschewski, B.: The real numbers in pointfree topology, Textos de Matemática Série B, No. 12, Departamento de Matemática da Universidade de Coimbra, 94 pp (1997)Google Scholar
  4. 4.
    Banaschewski, B., Pultr, A.: Booleanization. Cah. Topol. Géom. Differ. Catég. 37, 41–60 (1996)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cameron, D.E.: Maximal and minimal topologies. Trans. Am. Math. Soc. 160, 229–248 (1971)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cameron, D.E.: A class of maximal topologies. Pac. J. Math. 700, 229–248 (1977)MathSciNetGoogle Scholar
  7. 7.
    Dube, T.: When Boole commutes with Hewitt and Lindelöf. Appl. Categ. Struct. 25, 1097–1111 (2017)CrossRefGoogle Scholar
  8. 8.
    Ferreira, M.J., Picado, J., Pinto, S.M.: Remainders in pointfree topology. Topol. Appl. 245, 21–45 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gillman, L., Jerison, M.: Rings of Continuous Functions. Van Nostrand, Princeton (1960)CrossRefGoogle Scholar
  10. 10.
    Gutiérrez García, J., Kubiak, T.: A preservation result for completely regular locales. Topol. Appl. 168, 40–45 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    García, J. Gutiérrez, Kubiak, T., Picado, J.: On hereditary properties of extremally disconnected frames and normal frames. Topol. Appl. (to appear)Google Scholar
  12. 12.
    García, J.G., Picado, J.: On the parallel between normality and extremal disconnectedness. J. Pure Appl. Algebra 218, 784–803 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Isbell, J.R.: Graduation and dimension in locales. London Mathematical Society Lecture Notes Series 93, pp. 195–210. Cambridge University Press (1985)Google Scholar
  14. 14.
    Madden, J., Vermeer, J.: Lindelöf locales and realcompactness. Math. Proc. Camb. Philos. Soc. 99, 473–480 (1986)CrossRefGoogle Scholar
  15. 15.
    Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)zbMATHGoogle Scholar
  16. 16.
    Picado, J., Pultr, A.: Frames and Locales: Topology Without Points. Frontiers in Mathematics. Springer, Basel (2012)CrossRefGoogle Scholar
  17. 17.
    Picado, J., Pultr, A., Tozzi, A.: Joins of closed sublocales. Houst. J. Math. 45, 21–38 (2019)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Plewe, T.: Sublocale lattices. J. Pure Appl. Algebra 168, 309–326 (2002)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Raha, A.B.: Maximal topologies. J. Aust. Math. Soc. 15, 279–290 (1973)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Smythe, N., Wilkins, C.A.: Minimal Hausdorff and maximal compact spaces. J. Aust. Math. Soc. 3, 167–171 (1963)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Thomas, J.P.: Maximal connected topologies. J. Aust. Math. Soc. 8, 700–705 (1968)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Vermeulen, J.J.C.: Proper maps of locales. J. Pure Appl. Algebra 92, 79–107 (1994)MathSciNetCrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of South AfricaPretoriaSouth Africa

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