2-to-1 maps on hereditarily indecomposable continua

Heath, Jo

doi:10.1090/S0002-9947-1991-1028310-7

$2$-to-$1$ maps on hereditarily indecomposable continua
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by Jo Heath

Trans. Amer. Math. Soc. 328 (1991), 433-444

DOI: https://doi.org/10.1090/S0002-9947-1991-1028310-7

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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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Topologie

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