Abstract
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.
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Acknowledgements
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.
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Communicated by A. Constantin.
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Bella, A., Spadaro, S. On the cardinality of almost discretely Lindelöf spaces. Monatsh Math 186, 345–353 (2018). https://doi.org/10.1007/s00605-017-1112-4
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DOI: https://doi.org/10.1007/s00605-017-1112-4
Keywords
- Cardinal inequality
- Lindelöf space
- Arhangel’skii Theorem
- Elementary submodel
- Left-separated set
- Right-separated set
- Discrete set
- Free sequence