Hereditarily Lindelöf and hereditarily separable subspaces of function spaces

Abstract

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 $ω_1$. We prove that under $\mathrm{MA}(ω_1)$, if all finite powers of $X$ belong to NFS, then $C_p(X)^ω$ is ESLC; thus, for subspaces of $C_p(X)$, 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 $C_p(X)$ contains no $L$-spaces if $X$ is a Lindelöf $Σ$-space of countable tightness and $\mathrm{MA}(ω_1)$ holds, or if $X$ is any Lindelöf $Σ$-space and PFA holds. Under $\mathrm{MA}(ω_1)$, all iterated function spaces of Corson compact spaces are ESLC, and in ZFC the space $C_p(X,\{0,1\})$ is ESLC for every zero-dimensional Corson compact space $X$.