Lindelöf \(\Sigma\)-property of \(C_p(X)\) together with countable spread of \(X\) implies \(X\) is cosmic (Q2746304)

All spaces under consideration are assumed to be Tikhonov. A topological space \(X\) is called Lindelöf \(\Sigma\)-space if it is a continuous image of a space \(Y\) which can be mapped perfectly onto a second countable space. The author proves that if \(X\) has countable spread and \(C_p(X)\) is a Lindelöf \(\Sigma\)-space, then \(X\) has a countable network. This gives an answer to Problem 11 from [\textit{A. V. Arkhangel'skij}, Topology Appl. 74, No.~1-3, 83-90 (1996; Zbl 0865.54006)]. The author establishes that if \(X\) is a pseudocompact space with Souslin property, then every Lindelöf \(\Sigma\)-subspace of \(C_p(X)\) has a countable network. Also, he proves that if \(X\) is a Lindelöf \(\Sigma\)-space and \(\omega_1\) is a caliber of \(X\), then any Lindelöf \(\Sigma\)-subspace of \(C_p(X)\) has a countable network. The author shows that the same is true for a space with a dense subspace which is a continuous image of a product of separable spaces. At the end he poses ten open problems on this topic.