On finite-to-one open mappings (Q735659)

The author considers continuous surjective open finite-to-one mappings in the class of Hausdorff spaces. The following notion of a \textit{rigid space} is introduced and investigated. A space \(Y\) is called rigid if any finite-to-one mapping on \(Y\) is elementary. A mapping \(f:X\to Y\) is elementary if \(X\) can be represented as a topological sum \(\bigoplus_{i\leq k}X_i\) such that the restriction \(f|X_i\) is a homeomorphism of \(X_i\) on \(Y\). It is shown that the following spaces are rigid: topological sums of rigid spaces; zero-dimensional Lindelöf spaces; hereditarily disconnected locally compact paracompact spaces; the Stone-Čech compactifications of strongly zero-dimensional spaces; stably locally connected spaces; Euclidean spaces. Some examples of non-rigid spaces also are constructed. 6 open problems concerning rigid spaces are formulated at the end of the paper.