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Since there are L-spaces (provably in ZFC), under CH we have regular, hereditarily Lindelöf spaces with density continuum. However, I cannot find an example of such a space under not CH, nor a proof that no such space exists under axioms like MA or PFA. Shelah has a paper on the density of HL spaces, but it is somewhat opaque (I believe it is number 918 in his archive).

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    $\begingroup$ Ma does not exclude the existence of such a space. More precisely, using the method of [1] I think that I can prove that Con(ZFC + $2^{\omega}=\omega_2$ + MA + there is a hereditarily Lindelof space with density $\omega_2$.) [1]Soukup, L. Indestructible properties of S- and L-spaces. Topology Appl. 112 (2001), no. 3, 245–257. $\endgroup$
    – Lajos Soukup
    Commented Feb 25, 2023 at 15:02
  • $\begingroup$ Thank you for the answer and reference! I was working through the Abraham-Todorcevic Handbook article and couldn't find an answer $\endgroup$
    – GAW
    Commented Feb 27, 2023 at 1:30

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