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Weakly confluent, $ 2$-to-$ 1$ maps on hereditarily indecomposable continua


Author: Jo W. Heath
Journal: Proc. Amer. Math. Soc. 117 (1993), 569-573
MSC: Primary 54F15
DOI: https://doi.org/10.1090/S0002-9939-1993-1129879-1
MathSciNet review: 1129879
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Abstract: One of the most important open questions today in the study of exactly $ 2$-to-$ 1$ maps is whether or not such a map can be defined on the pseudoarc. We show that there is no weakly confluent $ 2$-to-$ 1$ map defined on the pseudoarc. More precisely, it is shown that any reduced weakly confluent $ 2$-to-$ 1$ map defined on a hereditarily indecomposable metric continuum must be a confluent local homeomorphism. It follows from this that if there is a weakly confluent $ 2$-to-$ 1$ map from a hereditarily indecomposable continuum $ X$ onto a continuum $ Y$, then neither $ X$ nor $ Y$ can be treelike.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1129879-1
Keywords: Weakly confluent, $ 2$-to-$ 1$ map
Article copyright: © Copyright 1993 American Mathematical Society

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