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.
Keywords: Lindelöf space; scattered space; \sigma-product; function space; P-space; exponentially separable space; product; functionally countable space; weakly exponentially separable space
DOI: DOI 10.14712/1213-7243.2022.021
AMS Subject Classification: 54G12 54G10 54C35 54D65