Hyperspaces and open monotone maps of hereditarily indecomposable continua

We prove the following theorems: Theorem 1. Let X be an n-dimensional hereditarily indecomposable continuum. Then there exist l-dimensional hereditarily indecomposable continua Y₁, Y₂,...) Y_n <nd monotone maps pi : X → Y₁ such that (p1,p2,...,pn) X → KI × YI × ... × Y_n is an embedding and the space C(X) of all subcontinua of X is embeddable in C(Yi) × C(V2) × ... × C(Y_n) by K ∈ C(X) ∈ (pi(K),p2(K),...,Pn(K)). Theorem 2. For every open monotone map if with non-trivial sufficiently small fibers on a finite dimensional hereditarily indecomposable continuum X with dim X > 2 there exists α l-dimensional subcontinuum Y ⊂ X such that dim φ(Y) = ∞ and the restriction of fp to Y is also monotone and open. The connection between these theorems and other results in Hyperspace theory is studied.