Hereditarily Lindelöf and hereditarily separable subspaces of function spaces

A space is ESLC if it contains neither an S-space nor an L-space, and NFS is the class of spaces containing no free sequences of length ω₁. We prove that under MA(ω₁), if all finite powers of X belong to NFS, then Cₚ(X)^ω is ESLC; thus, for subspaces of Cₚ(X), hereditary Lindelöfness, hereditary separability and countable spread coincide. This is derived from a dual form of a theorem of Todorčević. A Lindelöf $Σ$-space belongs to NFS if and only if its compact subspaces have countable tightness. Hence Cₚ(X) contains no L-spaces if X is a Lindelöf $Σ$-space of countable tightness and MA(ω₁) holds, or if X is any Lindelöf $Σ$-space and PFA holds. Under MA(ω₁), all iterated function spaces of Corson compact spaces are ESLC, and in ZFC the space Cₚ(X,\0,1\) is ESLC for every zero-dimensional Corson compact space X.