The space Cp(X) admits a dense exponentially separable subspace when X is metrizable

Abstract

We prove that $C_p\left(X\right)$ admits a dense exponentially separable for any metrizable space X and, answering a question in [16], we give an example of a pseudocompact ω-monolithic space such that $C_p\left(X\right)$ does not admit dense functionally countable subspaces. In a similar sense, solving consistently a problem in [11], we prove that $C_p\left(X\right)$ may not always contain a dense subspace of countable functional tightness. In other direction, and answering a question posed in [6], we characterize compact spaces for which their Alexandroff doubles have a Lindelöf $C_p$; we also give a short proof of a result in [19] about a consistent characterization of the Lindelöf property in $C_p$-spaces over Hattori spaces.