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Given a class M of mappings f between continua, near-M stands for the class of uniform limits of sequences of mappings from M. Let 2f and C(f) mean the induced mappings between hyperspaces. Relations are studied between the conditions: f∈ near-M, 2f∈ near-M and C(f)∈ near-M. A special attention is paid to the classes M of open and of monotone mappings.
For a given mapping f between continua we consider the induced mappings between the corresponding hyperspaces of closed subsets or of subcontinua. It is shown that if either of the two induced mappings is hereditarily weakly confluent (or hereditarily confluent, or hereditarily monotone, or atomic), then f is a homeomorphism, and consequently so are both the induced mappings. Similar results are obtained...
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