On spaces with point-countable cs-networks
A cover \({\mathcal P}\) of a space \(X\) is called a \(cs\)-network (resp. \(cs^*\)-network) if, for any open neighbourhood \(U\) of \(x\), and for any sequence \(L\) converging to \(x\), there exists \(P\in{\mathcal P}\) such that \(x\in P\subset U\), and \(P\) contains \(L\) eventuelly (resp. frequently). This paper obtains the results (A)\(\sim\)(D) below related to the following questions. \((\text{Q}_1)\) is whether every sequential space with a point-countable \(cs^*\)-network is a compact-covering quotient \(s\)-image of a metric space, and \((\text{Q}_2)\) is whether Fréchet spaces with point-countable \(cs^*\)-networks are preserverd by pseudo-open \(s\)-maps. \((\text{Q}_1)\) Is every quotient \(s\)-image of a metric space a compact-covering quotient \(s\)-image of a metric space? (\textit{E. Michael} and \textit{K. Nagami}'s question in [Proc. Am. Math. Soc. 37, 260-266 (1973; Zbl 0228.54008, Zbl 0245.54023)]). \((\text{Q}_2)\) Are pseudo-open \(s\)-images of metric spaces preserved by pseudo-open \(s\)-maps? (\textit{G. Gruenhage}, \textit{E. Michael} and \textit{Y. Tanaka}'s question in [Pac. J. Math. 113, 303-332 (1984; Zbl 0561.54016)]). \((\text{Q}_3)\) For quotient \(s\)-images \(X\) and \(Y\) of metric spaces, what is a necessary and sufficient condition for \(X\times Y\) to be a \(k\)-space? (\textit{Y. Tanaka}'s question in [Quest. Answers Gen. Topology 1, 36-50 (1983; Zbl 0545.54020)]). (A) \(X\) is a sequential space with a point-countable \(cs\)-network iff \(X\) is a compact-covering, sequence-covering, quotient and \(s\)-image of a metric space. (B) Every closed map from a space \(X\) with a point-countable \(k\)-network is compact-covering if each point of \(X\) is a \(G_\delta\)-set. (C) Every quotient \(s\)-image of a Fréchet space with a point-countable \(cs\)-network has a point-countable \(cs^*\)-network. (D) Let \(X\) and \(Y\) be sequential spaces with point-countable \(cs\)-networks. Then \(X\times Y\) is a \(k\)-space iff one of the following properties holds. (a) \(X\) and \(Y\) are first countable spaces; (b) \(X\) or \(Y\) is a locally compact space; (c) \(X\) and \(Y\) are locally \(k_\omega\)-spaces.
scientific article; zbMATH DE number 987911
scientific article