The one-point Lindelöfication of an uncountable discrete space can be surlindelöf (Q386363)

The author makes one of those simple but clever constructions to show that assuming the combinatorial principle stick, the one-point Lindelöfication of the discrete space of size \(\aleph_1\) is surlindelöf, that is, it is homeomorphic to some subspace of \(C_p(X)\), for some Lindelöf space \(X\). In the introduction of the paper, the author explains why it is important to know this fact about that space. The principle stick follows from both \(\mathsf{CH}\) and \(\clubsuit\); it implies \(\mathfrak p=\aleph_1\), so it is incompatible with \(\mathsf{MA}_{\omega_1}\). The main technique of the construction is to build a suitable subspace \(X_\mathcal A\) of the space \(C_p(\lambda D(\omega_1),2)\) with the help of the combinatorial principle; here \(\lambda D(\omega_1)\) stands for the one-point Lindelöfication of the discrete space of size \(\aleph_1\). The construction is in such a way that there is an uncountable subset \(C\) of \(\omega_1\) such that \(C\cup\{\infty\}\) with the topology generated by restrictions of functions in \(X_\mathcal A\) is homeomorphic to \(\lambda D(\omega_1)\). The paper ends with some open problems; for instance, the author asks whether \(\mathsf{MA}_{\omega_1}\) or even \(\mathsf{PFA}\) implies that \(\lambda D(\omega_1)\) is not a surlindelöf space.