OPENNESS OF INDUCED PROJECTIONS

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 2X denote the hyperspace of all nonempty closed subsets of X equipped with the Hausdor metric H dened by H(A;B) = maxfsupfd(a;B) : a 2 Ag; supfd(b; A) : b 2 Bgg (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 2X. 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) 2f: 2X! 2Y and C(f) : C(X)! C(Y) de ned by 2f(A) = f(A) for every A 2 2X and C(f)(A) = f(A) for every A 2 C(X): A mapping f: X! Y between spaces X and Y is said to be open provided the image of an open subset of the domain is open in the range. The following results concerning induced mappings for the class of open mappings are known (see [4, Theorem 4.3]; compare also [3, Theorem 3.2]). 1. Statement. Let a surjective mapping f: X! Y between continua X and Y be given. Consider the following conditions: (1.1) f: X! Y is open; (1.2) C(f) : C(X)! C(Y) is open; (1.3) 2f: 2X! 2Y is open. Then (1:1) and (1:3) are equivalent, and each of them is implied by (1:2). Received by the editors February 2, 1999