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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Hyperspaces and open monotone maps
of hereditarily indecomposable continua


Author: Michael Levin
Journal: Proc. Amer. Math. Soc. 125 (1997), 603-609
MSC (1991): Primary 54B20, 54F15, 54F45
MathSciNet review: 1389527
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Abstract: We prove the following theorems:

Theorem 1. Let $X$ be an $n$-dimensional hereditarily indecomposable continuum. Then there exist $1$-dimensional hereditarily indecomposable continua $Y_1,Y_2,...,Y_n$ and monotone maps $p_i :X \longrightarrow Y_i$ such that $(p_1,p_2,...,p_n) :X \longrightarrow Y_1 \times Y_2 \times ... \times Y_n$ is an embedding and the space ${\mathcal C}(X)$ of all subcontinua of $X$ is embeddable in ${\mathcal C}(Y_1) \times {\mathcal C}(Y_2) \times ... \times {\mathcal C}(Y_n)$ by $K \in {\mathcal C}(X) \longrightarrow (p_1(K),p_2(K),...,p_n(K))$.

Theorem 2. For every open monotone map $\varphi $ with non-trivial sufficiently small fibers on a finite dimensional hereditarily indecomposable continuum $X$ with $\dim X \geq 2$ there exists a $1$-dimensional subcontinuum $Y \subset X$ such that $\dim \varphi (Y) = \infty $ and the restriction of $\varphi $ to $Y$ is also monotone and open.

The connection between these theorems and other results in Hyperspace theory is studied.


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Additional Information

Michael Levin
Affiliation: Department of Mathematics, Haifa University, Mount Carmel, Haifa 31905, Israel
Address at time of publication: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350
Email: levin@mathcs2.haifa.ac.il, levin@math.washington.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03855-0
PII: S 0002-9939(97)03855-0
Keywords: Hyperspaces, hereditarily indecomposable continua, open monotone maps
Received by editor(s): January 1, 1995
Communicated by: James West
Article copyright: © Copyright 1997 American Mathematical Society