Limit properties of induced mappings
Given a mapping \(f:X\to Y\) between continua \(X\) and \(Y\), let \(2^f: 2^X\to 2^Y\) and \(C(f): C(X)\to C(Y)\) denote the corresponding induced mappings. Let \({\mathcal M}\) be a class of mappings between continua. A general problem which is related to a given mapping and to the two induced mappings is to find all interrelations between the following three statements: (1) \(f\in {\mathcal M}\), (2) \(C(f)\in {\mathcal M}\); and (3) \(2^f\in{\mathcal M}\). There are some papers in which particular results concerning this problem are shown for various classes \({\mathcal M}\) of mappings like open, monotone, confluent and some others. In this paper the authors discuss the problem concerning possible relations between conditions (1)--(3) from one side, and the corresponding conditions in which an admissible class near-\({\mathcal M}\), defined as the class of uniform limits of mappings belonging to \({\mathcal M}\), from the other. Special attention is paid to the classes \({\mathcal M}\) of open and of monotone mappings.
scientific article; zbMATH DE number 1395057