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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

$2$-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
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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.


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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