Abstract. 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 X denote the hyperspace of all nonempty closed subsets of X equipped with the Hausdorff metric H defined by