Question. Suppose that $X$ is a Lindelof space such that every point of $X$ is a $G_{\delta}$point. Then is it true that $X ≤ 2^{\omega}$?
3 Answers
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. 4566.
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''.
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.

4$\begingroup$ 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 ♦Jun 4, 2015 at 16:49

2$\begingroup$ Would you please say which part of the argument is wrong? $\endgroup$ Jun 5, 2015 at 5:20

$\begingroup$ For reference, Knight's paper is but a click away: journals.impan.pl/cgibin/doi?fm19413 $\endgroup$– David Roberts ♦Jun 6, 2015 at 0:31
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 (TallUsuba): Lévycollapse 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 $\leq \aleph_1$".