Reflecting Lindelöfness (Q1612233)

Must every Lindelöf space have a Lindelöf subspace of size \(\aleph_1\)? This paper provides a variety of partial answers to this question. Some of the results are: (a) For many classes of Hausdorff spaces the answer is ``yes'' under CH: these classes include sequential, locally separable, countable spread, countable tightness, \(k\)-space; without CH the answer is ``yes'' for \(k\)-spaces of uncountable tightness. (b) There is an uncountable Lindelöf \(T_1\) space with no Lindelöf subspace of size \(\aleph_1\). (c) Cons(\(2^{\aleph_0} = \aleph_2\) + ``yes for Hausdorff spaces of character \(\leq \aleph_1\)'') assuming Cons(there is a weakly compact cardinal). Editorial remark: In [\textit{R. de la Vega}, Fundam. Math. 247, No. 2, 165--170 (2019; Zbl 1433.54005)] a counter example is given to a claim given in the introduction of this paper concerning the countable chain condition (ccc).