Minimal uniformizability revisited in terms of normal sequence of covers (Q748233)

A topological space \((X,\tau)\) is said to be \textit{minimal uniformizable} (which coincides with the notion of \textit{minimal completely regular} defined by \textit{M. P. Berri} in [Trans. Am. Math. Soc. 108, 97--105 (1963; Zbl 0114.13902)]) if the only uniformizable topology over \(X\) strictly weaker than \(\tau\) is the indiscrete topology. For two covers \(\mathcal{U}\) and \(\mathcal{V}\) of \(X\), we say that \(\mathcal{U}\) \textit{star refines} \(\mathcal{V}\), or \(\mathcal{U}\) is a \textit{star refinement} of \(\mathcal{V}\), and we denote this by \(\mathcal{U} <^* \mathcal{V}\), if for each \(U \in \mathcal{U}\) there is a \(V \in \mathcal{V}\) such that \(\text{St}(U, \mathcal{U}) \subseteq V\), where \(\text{St}(U, \mathcal{U}) = \bigcup\{W \in \mathcal{U}: W \cap U \neq \emptyset\}\), while \(\mathcal{U}\) is said to be a \textit{refinement} of \(\mathcal{V}\), denoted by \(\mathcal{U} < \mathcal{V}\), if for each \(U \in \mathcal{U}\) there exists a \(V \in \mathcal{V}\) such that \(U \subseteq V\). Obviously if \(\mathcal{U} <^* \mathcal{V}\) then \(\mathcal{U} < \mathcal{V}\). A \textit{normal sequence of covers} is a sequence of open covers \(\mathcal{U}_1, \mathcal{U}_2, \ldots\) of \(X\) such that \(\mathcal{U}_{n + 1} <^* \mathcal{U}_n\) for every \(n \geqslant 1\), and a \textit{normal cover} is a cover which is \(\mathcal{U}_1\) in some normal sequence of covers. In the paper under review, the authors characterize minimal uniformizability in terms of normal sequences of covers (Theorem 3.4) and deduce a number of corollaries. For instance, it is shown in the paper that every minimal uniformizable, non-indiscrete topological space is pseudometrizable.