On inverse limits of weakly \(\bar\theta\)-refinable spaces (Q2704421)

\textit{K. Chiba} [Math. Jap. 35, No. 5, 959-970 (1990; Zbl 0731.54010)] discussed the inverse limits of some covering properties and normality. The purpose of this paper is to study the inverse limits of weakly \(\overline\theta\)-refinable spaces. A space \(X\) is weakly \(\overline \theta\)-refinable if every open cover of \(X\) has an open refinement \(\bigcup_{n \in\omega}{\mathcal U}_n\) such that (1) for each \(x\in X\) there is \(n\in \mathbb{N}\) such that \(0<\text{ord}(x,{\mathcal U}_n) <\omega\), and (2) \(\{\bigcup{\mathcal U}_n: n\in \omega\}\) is point-finite in \(X\). The main result is that, let \(X\) be the limit of an inverse system \(\{X_\alpha, \pi^\alpha_\beta, \Lambda\}\) and \(\lambda\) the cardinal number of \(\Lambda\), suppose each projection \(\pi_\alpha: X\to X_\alpha\) is an open and onto map and \(X\) is \(\lambda\)-paracompact, if each \(X_\alpha\) is normal and weakly \(\overline\theta\)-refinable, then \(X\) is normal and weakly \(\overline\theta\)-refinable. An analogous result for hereditarily \(\overline \theta\)-refinable spaces is obtained.