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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Concerning exactly $ (n,\,1)$ images of continua


Authors: Sam B. Nadler and L. E. Ward
Journal: Proc. Amer. Math. Soc. 87 (1983), 351-354
MSC: Primary 54F20; Secondary 54F50
DOI: https://doi.org/10.1090/S0002-9939-1983-0681847-3
MathSciNet review: 681847
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Abstract: A surjective mapping $ f:X \to Y$ is exactly $ (n,1)$ if $ {f^{ - 1}}(y)$ contains exactly $ n$ points for each $ y \in Y$. We show that if $ Y$ is a continuum such that each nondegenerate subcontinuum of $ Y$ has an endpoint, and if $ 2 \leqslant n < \infty $, then there is no exactly $ (n,1)$ mapping from any continuum onto $ Y$. However, if $ Y$ is a continuum which contains a nonunicoherent subcontinuum, then such an $ (n,1)$ mapping exists. Therefore, a Peano continuum is a dendrite if and only if for each $ n$ $ (2 \leqslant n < \infty )$ there is no exactly $ (n,1)$ mapping from any continuum onto $ Y$. We also show that for each positive integer $ n$ there is an exactly $ (n,1)$ mapping from the Hilbert cube onto itself.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0681847-3
Keywords: Exactly $ (n,1)$ mapping, continuum, dendrite, Hilbert cube
Article copyright: © Copyright 1983 American Mathematical Society