Monatshefte für Mathematik

, Volume 186, Issue 2, pp 345–353 | Cite as

On the cardinality of almost discretely Lindelöf spaces



A space is said to be almost discretely Lindelöf if every discrete subset can be covered by a Lindelöf subspace. Juhász et al. (Weakly linearly Lindelöf monotonically normal spaces are Lindelöf, preprint, arXiv:1610.04506) asked whether every almost discretely Lindelöf first-countable Hausdorff space has cardinality at most continuum. We prove that this is the case under \(2^{<{\mathfrak {c}}}={\mathfrak {c}}\) (which is a consequence of Martin’s Axiom, for example) and for Urysohn spaces in ZFC, thus improving a result by Juhász et al. (First-countable and almost discretely Lindelöf \(T_3\) spaces have cardinality at most continuum, preprint, arXiv:1612.06651). We conclude with a few related results and questions.


Cardinal inequality Lindelöf space Arhangel’skii Theorem Elementary submodel Left-separated set Right-separated set Discrete set Free sequence 

Mathematics Subject Classification

Primary 54A25 54D20 Secondary 54D35 54D10 54D55 



The first-named author was partially supported by a Grant of the Group GNSAGA of INDAM. The second-named author is grateful to FAPESP for financial support through postdoctoral Grant 2013/14640-1, Discrete sets and cardinal invariants in set-theoretic topology. Part of the research for the paper was carried out when he visited the first-named author at the University of Catania in December 2016. He thanks his colleagues there for the warm hospitality. The authors are grateful to Lajos Soukup for spotting an error in an earlier version of the paper.


  1. 1.
    Arhangel’skii, A.: On the cardinality of bicompacta satisfying the first axiom of countability. Soviet Math. Dokl. 10, 951–955 (1969)MATHGoogle Scholar
  2. 2.
    Balogh, Z.: On density and the number of \(G_\delta \) points in somewhat Lindelöf spaces. Topol. Proc. 27, 9–14 (2003)Google Scholar
  3. 3.
    Bell, M., Ginsburg, J., Woods, G.: Cardinal inequalities for topological spaces involving the weak Lindelöf number. Pacific J. Math. 79, 37–45 (1978)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bella, A., Cammaroto, F.: On the cardinality of Urysohn spaces. Can. Math. Bull. 31, 153–158 (1988)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dow, A.: An introduction to applications of elementary submodels to topology. Topol. Proc. 13(1), 17–72 (1988)MathSciNetMATHGoogle Scholar
  6. 6.
    Engelking, R.: General Topology. PWN, Warsaw (1977)MATHGoogle Scholar
  7. 7.
    Hodel, R.E.: Arhangelskii’s solution to Alexandroff’s problem. Topol. Appl. 153, 2199–2217 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Juhász, I.: Cardinal functions in topology—ten years later. In: Mathematical Centre tracts, vol. 123, Mathematisch Centrum, Amsterdam (1980)Google Scholar
  9. 9.
    Juhász, I., Soukup, L., Szentmiklóssy, Z.: First-countable and almost discretely Lindelöf \(T_3\) spaces have cardinality at most continuum, preprint, arXiv:1612.06651
  10. 10.
    Juhász, I., Tkachuk, V.V., Wilson, R.G.: Weakly linearly Lindelöf monotonically normal spaces are Lindelöf, preprint, arXiv:1610.04506
  11. 11.
    Kunen, K.: Set Theory, Studies in Logic, vol. 34. College Publications, London (2011)MATHGoogle Scholar
  12. 12.
    Veličko, N.: \(H\)-closed topological spaces, Math. Sb. (NS) 70 (1966), pp 98–112; Amer. Math. Soc. Transl. 78 (2) (1969), pp. 103–118Google Scholar

Copyright information

© Springer-Verlag GmbH Austria 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of CataniaCataniaItaly
  2. 2.Instituto de Matematica e Estatistica (IME-USP)Universidade de Sao PauloSão PauloBrazil

Personalised recommendations