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

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- by Sam B. Nadler and L. E. Ward
- Proc. Amer. Math. Soc.
**87**(1983), 351-354 - DOI: https://doi.org/10.1090/S0002-9939-1983-0681847-3
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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.## References

- Karol Borsuk and R. Molski,
*On a class of continuous mappings*, Fund. Math.**45**(1957), 84–98. MR**102063**, DOI 10.4064/fm-45-1-84-98 - T. A. Chapman,
*Lectures on Hilbert cube manifolds*, Regional Conference Series in Mathematics, No. 28, American Mathematical Society, Providence, R.I., 1976. Expository lectures from the CBMS Regional Conference held at Guilford College, October 11-15, 1975. MR**0423357**, DOI 10.1090/cbms/028 - Paul Civin,
*Two-to-one mappings of manifolds*, Duke Math. J.**10**(1943), 49–57. MR**8697** - Paul W. Gilbert,
*$n$-to-one mappings of linear graphs*, Duke Math. J.**9**(1942), 475–486. MR**7106** - O. G. Harrold Jr.,
*The non-existence of a certain type of continuous transformation*, Duke Math. J.**5**(1939), 789–793. MR**1358** - O. G. Harrold Jr.,
*Exactly $(k,1)$ transformations on connected linear graphs*, Amer. J. Math.**62**(1940), 823–834. MR**2554**, DOI 10.2307/2371492 - O. G. Harrold Jr.,
*Continua of finite sections*, Duke Math. J.**8**(1941), 682–688. MR**5336** - Venable Martin and J. H. Roberts,
*Two-to-one transformations on 2-manifolds*, Trans. Amer. Math. Soc.**49**(1941), 1–17. MR**4129**, DOI 10.1090/S0002-9947-1941-0004129-9 - J. Mioduszewski,
*On two-to-one continuous functions*, Rozprawy Mat.**24**(1961), 43. MR**145490** - J. H. Roberts,
*Two-to-one transformations*, Duke Math. J.**6**(1940), 256–262. MR**1923**, DOI 10.1215/S0012-7094-40-00620-2 - Gordon Thomas Whyburn,
*Analytic Topology*, American Mathematical Society Colloquium Publications, Vol. 28, American Mathematical Society, New York, 1942. MR**0007095**, DOI 10.1090/coll/028

## Bibliographic Information

- © Copyright 1983 American Mathematical Society
- 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