The structure of (exactly) 2-to-1 maps on metric compacta

Heath, Jo

doi:10.1090/S0002-9939-1990-1013970-1

The structure of (exactly) $2$-to-$1$ maps on metric compacta
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Proc. Amer. Math. Soc. 110 (1990), 549-555 Request permission

Abstract:

It is shown that the domain of a $2$-to-$1$ continuous map $f$ contains two disjoint open sets $V$ and ${V^ \wedge }$ such that $f\left ( V \right ) = f\left ( {{V^ \wedge }} \right )$ and $f \upharpoonright V$ is a homeomorphism from $V$ onto a dense open subset of the image of $f$. The restriction of $f$ to $V \cup {V^ \wedge }$ is the first "fold", and $f$ is shown to be the union of a finite or transfinite sequence of similar folds.

References

O. G. Harrold Jr., The non-existence of a certain type of continuous transformation, Duke Math. J. 5 (1939), 789–793. MR 1358
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
Jo Heath, $k$-to-$1$ functions between graphs with finitely many discontinuities, Proc. Amer. Math. Soc. 103 (1988), no. 2, 661–666. MR 943101, DOI 10.1090/S0002-9939-1988-0943101-6

Tree-like continua and exactly

-to-

functions

105

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