A shrinkability criterion for complete metric spaces (Q1292717)

The authors extend a special shrinking criterion, which was established by \textit{R. H. Bing} [Ann. Math., II. Ser. 65, 363-374 (1957; Zbl 0078.15201)] for locally compact metric spaces, to complete metric spaces. They call an upper semicontinuous decomposition \(G\) of a metric space \(X\) locally metrically shrinkable if for each \(\varepsilon>0\) and each \(g_0\in G\) there exists a homeomorphism \(h\) of \(X\) onto itself which is fixed outside the \(\varepsilon\)-neighborhood of \(g_0\), \(\text{diam} h(g_0) <\varepsilon\), and for any other \(g\in G\) either \(\text{diam} h(g) <\varepsilon\) or \(h(g)\) lies in the \(\varepsilon\)-neighborhood of \(g\). The main result indicates that every countable, locally metrically shrinkable decomposition \(G\) of a complete metric space \(X\) is shrinkable; that is, the natural decomposition map \(X\to X/G\) can be approximated, arbitrarily closely, by homeomorphisms.