Metrisable general resolutions (Q1612283)

This is the first of a two part paper, the second part of which is \textit{K. Richardson} and \textit{St. Watson} [Topology Appl. 122, 605-615 (2002; Zbl 1024.54007)]. The review covers both parts. Resolutions are ways of replacing points in topological spaces by whole spaces. In the first paper this is done by taking a family \(\{A_\alpha:\alpha\in I\}\) of subsets of a space \(X\) and continuous maps \(f_\alpha:X\setminus A_\alpha\to Y_\alpha\). The point \(x\) is replaced by the product \(Y_x=\prod\{Y_\alpha:x\in A_\alpha\}\) (or just \(\{x\}\) if \(x\notin\bigcup_\alpha A_\alpha\)). A typical basic open set in \(Z=\bigcup_x\{x\}\times Y_x\) looks like \(\pi^{-1}[U]\cap\bigcap_{\alpha\in J}\sigma_\alpha^{-1}[V_\alpha]\), where \(U\) is open in \(X\) and \(\pi:Z\to X\) is the obvious map, and where \(J\subseteq I\) is finite, \(V_\alpha\) is open in \(Y_\alpha\) and \(\sigma_\alpha:Z\to Y_\alpha\) is defined by \(\sigma_\alpha(x,y)=f_\alpha(x)\) if \(x\in A_\alpha\) and \(\sigma_\alpha(x,y)=y_\alpha\) if \(x\notin A_\alpha\). In the special resolutions of the second paper one has \(I=X\), \(A_x=\{x\}\) and so \(f_x:X\setminus\{x\}\to Y_x\). Both papers study when a resolution is metrizable and when it is not. This largely depends on the set \(\Lambda=\bigcup\{A_\alpha:|Y_\alpha|>1\}\). If \(X\) is metrizable, \(\{A_\alpha:|Y_\alpha|>1\}\) is a point-countable family of closed sets and each \(Y_\alpha\) has a \(\sigma\)-locally finite base then \(Z\) has a \(\sigma\)-locally finite base provided \(\Lambda\) is \(F_\sigma\)-discrete, i.e., a countable union of closed discrete subsets. Both papers establish (with roughly the same proofs) that \(Z\) is not metrizable if \(\Lambda\) is not \(F_\sigma\)-discrete and, in the first paper: each \(Y_\alpha\) is compact and \(\{A_\alpha:\alpha\in I\}\) is a point-finite family of closed sets; or in the second paper: each \(Y_x\) has a compact boundary in \(Z\). The proof boils down to showing that if \(Z\) were metrizable then \(\{x: \operatorname {diam}Y_x\geq\varepsilon\}\) must be closed and discrete. In both papers this yields characterizations of metrizability of the resolution. The second paper goes on to show that some (non-metrizable) spaces have metrizable (and even discrete) resolutions and finishes with some conditions that ensure (hereditary) normality of resolutions.