$2$-to-$1$ maps on hereditarily indecomposable continua
Author:
Jo Heath
Journal:
Trans. Amer. Math. Soc. 328 (1991), 433-444
MSC:
Primary 54C10; Secondary 54F15
DOI:
https://doi.org/10.1090/S0002-9947-1991-1028310-7
MathSciNet review:
1028310
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Suppose $f$ is a $2{\text {-to-}}1$ continuous map from the hereditarily indecomposable continuum $X$ onto a continuum $Y$. In order for it to be the case that each proper subcontinuum $C$ in $Y$ has as its preimage two disjoint continua each of which $f$ maps homeomorphically onto $C$, it is obviously necessary that $f$ satisfy the condition that each nondense connected subset of $Y$ has disconnected preimage. We show that this condition is also sufficient, and thus any $2{\text {-to-}}1$ continuous map from a hereditarily indecomposable continuum satisfying this condition must be confluent and have an image that is hereditarily indecomposable.
- H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 (1967), 241–249. MR 220249, DOI https://doi.org/10.4064/fm-60-3-241-249
- Jo Heath, Tree-like continua and exactly $k$-to-$1$ functions, Proc. Amer. Math. Soc. 105 (1989), no. 3, 765–772. MR 936775, DOI https://doi.org/10.1090/S0002-9939-1989-0936775-8
- 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 https://doi.org/10.1090/S0002-9939-1986-0854049-8
- Jo Heath, The structure of (exactly) $2$-to-$1$ maps on metric compacta, Proc. Amer. Math. Soc. 110 (1990), no. 2, 549–555. MR 1013970, DOI https://doi.org/10.1090/S0002-9939-1990-1013970-1 C. Kuratowski, Topologie. I, II, PWN, Warsaw and Wroclaw, 1948, 1950.
- T. Bruce McLean, Confluent images of tree-like curves are tree-like, Duke Math. J. 39 (1972), 465–473. MR 305372
- J. Mioduszewski, On two-to-one continuous functions, Rozprawy Mat. 24 (1961), 43. MR 145490
- Ira Rosenholtz, Open maps of chainable continua, Proc. Amer. Math. Soc. 42 (1974), 258–264. MR 331346, DOI https://doi.org/10.1090/S0002-9939-1974-0331346-8
Retrieve articles in Transactions of the American Mathematical Society with MSC: 54C10, 54F15
Retrieve articles in all journals with MSC: 54C10, 54F15
Additional Information
Keywords:
Hereditarily indecomposable continua,
<!– MATH $k{\text {-to-}}1$ –> <IMG WIDTH="58" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$k{\text {-to-}}1$"> function,
<!– MATH $2{\text {-to-}}1$ –> <IMG WIDTH="57" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="$2{\text {-to-}}1$"> map,
<!– MATH $k{\text {-to-}}1$ –> <IMG WIDTH="58" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img23.gif" ALT="$k{\text {-to-}}1$"> map,
<!– MATH $2{\text {-to-}}1$ –> <IMG WIDTH="57" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$2{\text {-to-}}1$"> function
Article copyright:
© Copyright 1991
American Mathematical Society
