A remark on Newman's theorem (Q1921842)

Newman's theorem says that for a topological manifold \(M\) endowed with a metric \(d\) there is an \(\varepsilon = \varepsilon (M,d)>0\) such that for any finite group \(G\) and any non-trivial action of \(G\) on \(M\) by homeomorphisms there is at least one orbit of diameter \(\geq \varepsilon\). This was generalized by Chernavskij to continuous partitions of \(M\) induced by pseudosubmersions. Here a pseudosubmersion is a continuous surjective open and closed mapping \(f:X\to Y\) between topological spaces which is not a homeomorphism and for which every fibre \(f^{-1} (y),\;y\in Y\) is finite. The authors prove the following generalization: Let \((X,d)\) be a locally compact metric space which is a union of a finite number of open subsets which are topological manifolds. Then there is a positive number \(\varepsilon=\varepsilon (X,d)\) such that for every pseudosubmersion defined on \(X\) there is a fibre with diameter \(\geq\varepsilon\).