Hyperspaces and open monotone maps of hereditarily indecomposable continua

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.