A weakening of submetrizability and properties of spaces (Q2800380)

A topological space \(X\) is \textit{weakly jointly compact-metrizable}, briefly a wJCM-space, if there exists a metric \(d\) on \(X\) which metrizes all compact metrizable subspaces of \(X\). Obviously, JCM-spaces (introduced in [\textit{A. V. Arhangel'skii} et al., Topology Appl. 169, 2--15 (2014; Zbl 1294.54005)]), are weakly jointly compact-metrizable. The paper examines this newly introduced concept of wJCM-space. A sample result, see Corollary 3.2, says: Let \(f:X\to Y\) be a perfect mapping of a wJCM-space \(X\) onto a metrizable space \(Y\). If the fibres \(f^{-1}(y)\), where \(y\in Y\), are metrizable, then \(X\) is metrizable.