On closed inverse images of base-paracompact spaces (Q875193)

A Hausdorff space \(X\) is called base-paracompact if \(X\) has a base \(\mathcal B\) with \(| \mathcal B| =w(X)\) (the weight of \(X\)) such that every open cover of \(X\) has a locally finite refinement \(\mathcal B^\prime\) with \(\mathcal B^\prime\subset\mathcal B\). This notion was introduced by \textit{J. E. Porter} [Topology Appl. 128, 145--156 (2003; Zbl 1043.54010)]. Porter proved that base-paracompactness is inversely preserved by perfect maps. \textit{L. Wu} [Surveys in inverse images of covering properties, 2005 International General Topology Symposium, Zhangshou Teachers College, Zhangzhou, P.R. China] raised the question: Is base-paracompactness inversely preserved by closed Lindelöf maps? A map \(f:X\rightarrow Y\) is called Lindelöf provided \(f^{-1}(y)\) is Lindelöf for all \(y\in Y\). The author gives some partial answers to Wu's Question. The author defines a map \(f:X\rightarrow Y\) to be a base-paracompact mapping if there exists a base \(\mathcal B\) for \(X\) with \(| \mathcal B| =w(X)\) such that for every \(y\in Y\) and every family \(\mathcal U\) of open subset of \(X\) which covers \(f^{-1}(y)\), there exists an open neighborhood \(O_y\) of \(y\) and a partial refinement \(\mathcal B_y\) of \(\mathcal U\) where \(\mathcal B_y\subset \mathcal B\) such that \(f^{-1}(O_y)\subset \bigcup\mathcal B_y\) and \(\mathcal B_y\) is locally finite in \(X\) at every point of \(f^{-1}(O_y)\). Theorem 2.1: If \(f:X\rightarrow Y\) is a base-paracompact mapping, \(X\) is regular and \(w(X)\geq w(Y)\), then if \(Y\) is a base-paracompact space, then so is \(X\). Both perfect maps and (if the domain is regular) closed Lindelöf maps are base-paracompact mappings. Corollary 3.2: If \(f:X\rightarrow Y\) is a closed Lindelöf map, \(X\) is regular and \(w(X)\geq w(Y)\) then if \(Y\) is base-paracompact, so is \(X\). The author gives an example to show that ``\(X\) is regular'' cannot be weakened to ``\(X\) is Hausdorff'' in the statement of the Corollary. Whether or not ``\(w(X)\geq w(Y)\)'' can be deleted is not completely solved, but the author proves (but makes a weaker statement) that in the context of the the hypothesis of the Corollary, ``\(w(X)\geq w(Y)\)'' can be weakened to the cardinal arithmetic statement ``\(w(X)^\omega=w(X)\)''.