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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ 2$-to-$ 1$ maps on hereditarily indecomposable continua

Author: Jo Heath
Journal: Trans. Amer. Math. Soc. 328 (1991), 433-444
MSC: Primary 54C10; Secondary 54F15
MathSciNet review: 1028310
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Abstract: Suppose $ f$ is a $ 2{\text{-to-}}1$ continuous map from the hereditarily indecomposable continuum $ X$ onto a continuum $ Y$. In order for it to be the case that each proper subcontinuum $ C$ in $ Y$ has as its preimage two disjoint continua each of which $ f$ maps homeomorphically onto $ C$, it is obviously necessary that $ f$ satisfy the condition that each nondense connected subset of $ Y$ has disconnected preimage. We show that this condition is also sufficient, and thus any $ 2{\text{-to-}}1$ continuous map from a hereditarily indecomposable continuum satisfying this condition must be confluent and have an image that is hereditarily indecomposable.

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Keywords: Hereditarily indecomposable continua, $ k{\text{-to-}}1$ function, $ 2{\text{-to-}}1$ map, $ k{\text{-to-}}1$ map, $ 2{\text{-to-}}1$ function
Article copyright: © Copyright 1991 American Mathematical Society