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 $\left|Y\right|\le 2^{{\omega }_1}$ is scattered, then $X$ is functionally countable; if every $Y\subset X$ with $\left|Y\right|\le 2^{𝔠}$ is scattered, then...