A study of \(\gamma \)-metrizable spaces (Q2215657)

Let \(X\) be a topological space and let \(\gamma = \{X_i \colon i \in I\}\) be an open cover of \(X\). The space \(X\) is said to be \(\gamma\)-metrizable if there is a metric \(d\) on \(X\) such that, for every \(i \in I\), the subspace topology on \(X_i\) is metrizable by restriction \(d|_{X_i \times X_i}\). This concept was introduced by \textit{M. A. Al Shumrani} [Topology Appl. 264, 489--492 (2019; Zbl 1420.54044)]. The authors show that Theorems 2.1 and 2.4 of Al Shumrani's paper are false. In particular, an example of a regular \(\gamma\)-metrizable space is constructed. Some properties of \(\gamma\)-metrizable space are also discussed.