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 ^<c= 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 T₃ spaces have cardinality at most continuum, preprint, arXiv:1612.06651). We conclude with a few related results and questions.