Functional countability is preserved by some products
In this paper all spaces are Tychonoff. A space \(X\) is functionally countable if every real continuous image of \(X\) is countable. \(X\) is exponentially separable if for every family \(\mathcal{F}\) of closed subsets of \(X\), there is a countable subset of \(X\) which meets every non-empty intersection of members of \(\mathcal{F}\). From the author's paper [Commentat. Math. Univ. Carol. 59, No. 3, 399--409 (2018; Zbl 1424.54071)] an exponentially separable space is functionally countable, a scattered Lindelöf space is exponentially separable and a compact functionally countable space is scattered. In the paper under review, the author examines the behaviour of functional countability under products. The main results are as follows: If \(X\) is functionally countable and \(K\) is a scattered Corson compact space, then \(X \times K\) is functionally countable. If a functionally countable space \(X\) is weakly Lindelöf or separable or has the Souslin property, then \(X \times L\) is functionally countable for any scattered Lindelöf space \(L\). A pseudocompact product of finitely many functionally countable spaces is functionally countable. The product of finitely many functionally countable, countably compact spaces is functionally countable. The product of two exponentially separable, countably compact spaces is exponentially separable provided one of them is sequential.\par The paper concludes with a list of ten open problems.
scientific article; zbMATH DE number 7614064
scientific article