Exponential separability is preserved by some products

Tkachuk, Vladimir V.

Commentationes Mathematicae Universitatis Carolinae, Tome 63 (2022) no. 3, pp. 385-395.

Résumé

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.

Détail

DOI : 10.14712/1213-7243.2022.021

Classification : 54C35, 54D65, 54G10, 54G12
Keywords: Lindelöf space; scattered space; $\sigma$-product; function space; $P$-space; exponentially separable space; product; functionally countable space; weakly exponentially separable space