Exponential separability is preserved by some products

Abstract

We show that exponential separability is an inverse invariant of closed maps with countably compact exponentially separable fibers. This implies that it is preserved by products with a scattered compact factor and in the products of sequential countably compact spaces. We also provide an example of a $\sigma$-compact crowded space in which all countable subspaces are scattered. If $X$ is a Lindelöf space and every $Y\subset X$ with $|Y|\leq 2^{\omega_1}$ is scattered, then $X$ is functionally countable; if every $Y\subset X$ with $|Y|\leq 2^{\mathfrak{c}} $ is scattered, then $X$ is exponentially separable. A Lindelöf $\Sigma$-space $X$ must be exponentially separable provided that every $Y\subset X$ with $|Y|\leq {\mathfrak{c}}$ is scattered. Under the Luzin axiom ($2^{\omega_1}>{\mathfrak{c}} $) we characterize weak exponential separability of $C_p(X,[0,1])$ for any metrizable space $X$. Our results solve several published open questions.