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Limit properties of induced mappings. (English) Zbl 0994.54035

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.
Reviewer: Shou Lin (Fujian)

MSC:

54F15 Continua and generalizations
54B20 Hyperspaces in general topology
54E40 Special maps on metric spaces
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