On resolutions of generalized metric spaces (Q1763619)

Let \(X\) be a topological space. For each point \(x\in X\), let \(Y_x\) be a space and \(f_x:X\setminus\{x\}\rightarrow Y_x\) be a continuous map. The resolution \(Z\) of \(X\) at each point \(x\in X\) into \(Y_x\) by the mappings \(f_x\) is the space \(Z=\bigcup\{\{x\}\times Y_x:x\in X\}\) having as a base all sets of the form \(U\otimes V=(\{x\}\times V)\cup\bigcup\{\{x^\prime\}\times Y_{x^\prime}:x^\prime\in U\cap f_x^{-1}(V)\}\}\) where \(U\) is open in \(X\) and \(V\) is open in \(Y_x\). Resolutions were introduced by \textit{V. V. Fedorčuk} [Soviet Math. Dokl. 9 (5) 1148--1150 (1968; Zbl 0186.27003)]. \textit{S. Watson} [Recent Progress in General Topology, North-Holland, Amsterdam (1992; Zbl 0803.54001), Chapter 20] wrote an extensive article about resolutions showing that they unify many disparate constructions in topology. There has been significant activity towards determining properties of resolutions. For instance, Fedorchuk proved that if \(X\) is compact Hausdorff and \(Y_x\) is compact Hausdorff for all \(x\in X\), then the resolution \(Z\) is compact Hausdorff. \textit{K. Richardson} [Topology Appl. 122, No. 3, 593--603 (2002; Zbl 1024.54007)], and \textit{K. Richardson} and \textit{S. Watson} [Topology Appl. 21, 51--58 (1985; Zbl 1024.54006)] considered (among other things) metrizable and normal resolutions. In the present paper the authors consider several generalized metric properties such as \(G_\delta\)-diagonal, developable spaces, and \(M_0\), \(M_1\) and \(M_3\) spaces. Most of their results depend on the set \(\Lambda=\{x\in X:| Y_x| >1\}\) which played a role in the papers by Richardson. We state the main results from the present paper: (1) If \(X\) is has a \(G_\delta\)-diagonal and \(Y_x\) has a \(G_\delta\)-diagonal for all \(x\in X\) and \( \Lambda\) is a countable union of closed discrete subsets of \(X\), then the resolution \(Z\) has a \(G_\delta\)-diagonal. (2) If \(X\) is a developable space and \(Y_x\) is a compact metrizable space for all \(x\in X\), then the resolution \(Z\) is developable if and only if \(\Lambda\) is a countable union of closed discrete subsets of \(X\). (3) If \(X\) is an \(M_0\)-space and \(Y_x\) is an \(M_0\)-space for all \(x\in X\) and \( \Lambda\) is a countable union of closed discrete subsets of \(X\), then the resolution \(Z\) is an \(M_0\)-space. (4) If \(X\) is an \(M_3\)-space and \(Y_x\) is an \(M_3\)-space for all \(x\in X\) and \(\Lambda\) is a countable union of closed discrete subsets of \(X\), then the resolution \(Z\) is an \(M_3\)-space.