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Question. Suppose that $X$ is a Lindelof space such that every point of $X$ is a $G_{\delta}$-point. Then is it true that $|X| ≤ 2^{\omega}$?

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There are consistency results due to Shelah and Gorelic for the negative answer. Shelah has proved the consistency of $GCH+$there exists a Lindelof space with points $G_\delta$ of size $\aleph_2.$ See Shelah's paper ''On some problems in general topology''. Gorelic has extended this result to get $CH+2^{\aleph_1}$ arbitrary large+ there is such a space of size $2^{\aleph_1}$.

It is also consistent with $GCH$ that there exists a Lindelof space with points $G_\delta$ of size $\aleph_\omega.$ See R. Knight's paper ''A topological application of flat morasses'', Fundamenta Mathematicae, 194 (2007) pp. 45-66.

On the other hand Shelah has proved the consistency of $CH+$ there is no Lindelof space with points $G_\delta$ of size $\aleph_2.$ See Shelah's paper ''On some problems in general topology''.

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Unfortunately, the construction of a Lindelöf point $G_\delta$ space of size $\aleph_\omega$ in "A topological application of flat morasses" contains an error which seems to be incorrigible.

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    $\begingroup$ This seems to be a comment on Mohammad Golshani's answer, yes? Would you like to tell us what the error is? $\endgroup$
    – Todd Trimble
    Commented Jun 4, 2015 at 16:49
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    $\begingroup$ Would you please say which part of the argument is wrong? $\endgroup$
    – Mohammad Golshani
    Commented Jun 5, 2015 at 5:20
  • $\begingroup$ For reference, Knight's paper is but a click away: journals.impan.pl/cgi-bin/doi?fm194-1-3 $\endgroup$
    – David Roberts ♦
    Commented Jun 6, 2015 at 0:31
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The consistency of this statement is Problem 1 in these slides by F. Tall, where the problem is attributed to Arhangelski. Some partial results already mentioned in Mohammad Golshani´s answer are given. The following is also mentioned:

Theorem (Tall-Usuba): Lévy-collapse a weak compact and then add $\aleph_3$ Cohen subsets of $\omega_1$. Then there are no counterexamples of size $\aleph_2$, even relaxing "points $G_\delta$" to "pseudocharacter $\leq \aleph_1$".

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