Sequential order of product of Fréchet spaces
For a subset \(A\) of a space, \([A ]^{\text{seq}}\) denotes the sequential closure of \(X\), i.e., the set of limits of convergent sequences in \(A\). For a sequential space \(X\), we define \(\text{so} (X)\), the sequential order of \(X\), by \(\text{so} (X)= \min \{\alpha\in \omega_1 + 1: \text{cl } A=[ A]^\alpha\) for every \(A\subset X\}\), where \([A ]^0 =A\), \([A ]^{\alpha+ 1}= [[ A]^\alpha ]^{\text{seq}}\), and \([A ]^\alpha= \bigcup \{[ A]^\beta: \beta< \alpha\}\) for a limit \(\alpha\). A space \(X\) is strongly Fréchet if whenever \(x\in \text{cl } A_n\) \((n\in \mathbb{N})\), here \(A_{n+1} \subset A_n\), there exists a sequence \(\{x_n: n\in \mathbb{N}\}\) converging to \(x\) with \(x_n\in A_n\). In this paper, assuming CH, for any given \(\alpha\leq \omega_1\) there exist strongly Fréchet spaces \(X\) and \(Y\) such that \(X \times Y\) is sequential, and \(\text{so} (X\times Y)= \alpha\), thus answering a problem in the authors' previous paper [ibid. 65, 271-285 (1995; Zbl 0837.54021)].
scientific article; zbMATH DE number 896619
scientific article