On inverse limits of proposition \(\underline\theta\)-refinable spaces (Q2752628)

\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 \(\underline\theta\)-refinable spaces. A space \(X\) is \(\underline\theta\)-refinable if every directed open cover of \(X\) has a point-star refinement sequence of open covers of space \(X\). 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 \(\underline\theta\)-refinable, then \(X\) is normal and \(\underline\theta\)-refinable. An analogous result for hereditarily \(\underline\theta\)-refinable spaces is obtained.