On Lindelöf sets of continuous functions (Q1892172)

This paper studies properties of Lindelöf subspaces of the space \(C_ p(X)\) of continuous real-valued functions on a separable compact space \(X\). The main theorem is that, under \(\text{MA} + \neg \text{CH}\), any such subspace of \(C_ p (X)\) having every finite power Lindelöf must have a countable network. This result is related to theorems of Arkhangel'skij and Uspenskij, Sipacheva, and Reznichenko. The author applies this result to a construction in a prior paper to show that, under \(\text{MA} + \neg \text{CH}\), there exists a separable \(\sigma\)- compact space \(X\) such that \(C_ p (X)\) is not homeomorphic to a subspace of \(C_ p(K)\) for any separable compact space \(K\).