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2-to-1 maps with hereditarily indecomposable images. (English) Zbl 0738.54012

The author shows — in one of the final results of the paper — that exactly the 2-to-1 continuous image of the pseudoarc contains a subcontinuum which has a decomposition into two proper subcontinua such that one of these summands differs from the other by a connected set having connected inverse. It follows, in particular, that there is no exactly 2-to-1 continuous map from the pseudoarc onto a hereditarily indecomposable continuum. This non-existence holds, however — as is shown in the paper — also for maps from arbitrary treelike continua. A tool of the proofs is the authors lemma from the paper “\(K\)-to-1 maps on hereditarily indecomposable metric continua”, to appear in Trans. Am. Math. Soc., and a theorem by T. Bruce McLean [Duke Math. J. 39, 465-473 (1972; Zbl 0252.54020)], concerning confluent images of treelike continua.

MSC:

54F15 Continua and generalizations
54C10 Special maps on topological spaces (open, closed, perfect, etc.)

Citations:

Zbl 0252.54020
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