On \(s\)-images of metric spaces
Let \(X\) be a space, and \({\mathcal P}\) a cover of \(X\). \({\mathcal P}\) is said to be a \(k\)-network for \(X\) if \(K\subset U\) with \(K\) compact and \(U\) open, then \(K\subset\bigcup {\mathcal P}'\subset U\) for some finite subset \({\mathcal P}'\) of \({\mathcal P}\). If \({\mathcal P}\) is a \(k\)-network for \(X\), then \({\mathcal P}\) is a closed (countably compact, compact) \(k\)-network if each element of \({\mathcal P}\) is closed (countably compact, compact) in \(X\). Every quotient \(s\)-image of a metric space, or every closed image of a metric space has a point-countable \(k\)-network. \textit{S. Lin} [Topol. Proc. 20, 185-190 (1995; Zbl 0869.54025)] proved that for a regular and \(T_1\)-space \(X\), if \(X\) has a point-countable closed \(k\)-network, then \(X\) has a point-countable countably compact \(k\)-network if and only if every first countable closed subset of \(X\) is locally compact. The following question was posed: Suppose a space \(X\) has a point-countable closed \(k\)-network. Is \(X\) a space with a point-countable compact \(k\)-network if every first countable closed subspace of \(X\) is locally compact?NEWLINENEWLINENEWLINEIn the paper under review the author discusses some relations of the \(s\)-images of metric spaces and the spaces with a point-countable closed \(k\)-network, and constructs an example which answers negatively to the above question by showing that there is a regular \(T_1\) countably compact space \(Y\) such that \(Y\) has a point-countable closed \(k\)-network and every first countable closed subspace of \(Y\) is compact, but \(Y\) has no point-countable compact \(k\)-network.
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