Remarks on spaces with special type of \(k\)-networks (Q1375759)

A cover \({\mathcal P}\) of a space is a \(k\)-network if, for any compact subset \(K\) and any open subset \(U\) with \(K\subset U\), there exists a finite \({\mathcal P}^* \subset {\mathcal C}\) such that \(K\subset \bigcup {\mathcal P}^* \subset U\). This paper gives negative answers to the following questions posed by \textit{Y. Ikeda} and \textit{Y. Tanaka} [Topol. Proc. 18, 107-132 (1993; Zbl 0829.54018)] by means of subspaces of the Stone-Čech compactifications of discrete spaces. These questions have positive answers if \(X\) is a \(k\)-space, and so on. Spaces are completely regular and \(T_1\). (a) Does every closed image of a space \(X\) with a star-countable \(k\)-network have a (star-countable \(k\)-network, or) point-countable \(k\)-network? (b) Is every space \(X\) with a locally countable \(k\)-network a \((\sigma\)-space, or) space in which every closed subset is a \(G_\delta\)-set?