There is no exactly 𝑘-to-1 function from any continuum onto [0,1], or any dendrite, with only finitely many discontinuities

Heath, Jo W.

doi:10.1090/S0002-9947-1988-0927692-1

There is no exactly $k$-to-$1$ function from any continuum onto $[0,1]$, or any dendrite, with only finitely many discontinuities
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Trans. Amer. Math. Soc. 306 (1988), 293-305 Request permission

Abstract:

Katsuura and Kellum recently proved [8] that any (exactly) $k$-to$1$ function from $[0, 1]$ onto $[0, 1]$ must have infinitely many discontinuities, and they asked if the theorem remains true if the domain is any (compact metric) continuum. The result in this paper, that any (exactly) $k$-to-$1$ function from a continuum onto any dendrite has finitely many discontinuities, answers their question in the affirmative.

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
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
Jo Heath, Every exactly $2$-to-$1$ function on the reals has an infinite set of discontinuities, Proc. Amer. Math. Soc. 98 (1986), no. 2, 369–373. MR 854049, DOI 10.1090/S0002-9939-1986-0854049-8
Jo W. Heath, $K$-to-$1$ functions on arcs for $K$ even, Proc. Amer. Math. Soc. 101 (1987), no. 2, 387–391. MR 902560, DOI 10.1090/S0002-9939-1987-0902560-4
Hidefumi Katsuura and Kenneth R. Kellum, $k$-to-$1$ functions on an arc, Proc. Amer. Math. Soc. 101 (1987), no. 4, 629–633. MR 911022, DOI 10.1090/S0002-9939-1987-0911022-X
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
Sam B. Nadler Jr. and L. E. Ward Jr., Concerning exactly $(n,\,1)$ images of continua, Proc. Amer. Math. Soc. 87 (1983), no. 2, 351–354. MR 681847, DOI 10.1090/S0002-9939-1983-0681847-3
J. H. Roberts, Two-to-one transformations, Duke Math. J. 6 (1940), 256–262. MR 1923, DOI 10.1215/S0012-7094-40-00620-2