Openness of Induced Projections

For continua X and Y it is shown that if the projection f : X × Y → X has its induced mapping C(f) open, then X is C∗-smooth. As a corollary, a characterization of dendrites in these terms is obtained. All spaces considered in this paper are assumed to be metric. A mapping means a continuous function. To exclude some trivial statements we assume that all considered mappings are not constant. A continuum means a compact connected space. Given a continuum X with a metric d, we let 2 denote the hyperspace of all nonempty closed subsets of X equipped with the Hausdorff metric H defined by H(A,B) = max{sup{d(a,B) : a ∈ A}, sup{d(b, A) : b ∈ B}} (see e.g. [6, (0.1), p. 1, and (0.12), p. 10]). Further, we denote by C(X) the hyperspace of all subcontinua of X , i.e., of all connected elements of 2 . The reader is referred to Nadler’s book [6] for needed information on the structure of hyperspaces. Given a mapping f : X → Y between continua X and Y , we consider mappings (called the induced ones) 2 : 2 → 2 and C(f) : C(X)→ C(Y )