Let \(\kappa\) be an ordinal number. A mapping \( \langle x_{\xi}:\xi\in \kappa \rangle\) with values in a topological space \(X\) is called a chain of length \(\kappa\). We say that a chain \( \langle x_{\xi}:\xi\in \kappa \rangle \) converges to a point \(x\in X\) if for every neighborhood \(U\) of \(x\) there is some \(\alpha < \kappa\) such that \(x_{\xi}\in U\) for all \(\alpha\leq \xi < \kappa\). A topological space \(X\) is said to be pseudoradial if each of its nonclosed subsets \(A\) has a chain convergent to a point outside of \(A\). The chain character of a pseudoradial space \(X\), denoted by \(\sigma_{c}(X)\), is the smallest cardinal \(\kappa\) such that for any nonclosed set \(A\subseteq X\) there exist a point \(x\in cl_{X}A\setminus A\) and a chain \( \langle x_{\xi}:\xi\in \lambda \rangle \) of length \(\lambda \leq \kappa\) which converges to \(x\).
A major question in the study of pseudoradial spaces is whether the product of two compact pseudoradial spaces is pseudoradial. This question has been approached in many papers (some recent results can be consulted in [A. Bella, Proc. Am. Math. Soc. 119, 637-649 (1993; Zbl 0808.54006), Topology Appl. 70, 113-123 (1996; Zbl 0862.54022) and A. Bella, A. Dow and G. Tironi, Topology Appl. 111, 71-80 (2001; Zbl 0970.54035)]) and remains still open. In the paper under review, the authors address this interesting problem. The main result of the paper is the following: If \(X\) is a compact pseudoradial Hausdorff space with \(\sigma_{c}=\omega_{1}\) and \(Y\) is a compact, sequentially compact Hausdorff space such that each nonempty closed subset \(F\subseteq Y\) contains a point \(y\in Y\) whose character is less than or equal to \(\omega_{1}\), then \(X\times Y\) is pseudoradial. The authors also observe that the condition on the space \(Y\) is not satisfied by all pseudoradial spaces.