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It is shown that for locally connected continuum X if the induced mapping C(f) : C(X) → C(Y ) is open, then f is monotone. As a corollary it follows that if the continuum X is hereditarily locally connected and C(f) is open, then f is a homeomorphism. An example is given to show that local connectedness is essential in the result. All spaces considered in this paper are assumed to be metric. A mapping means a continuous function. We denote by N the set of all positive integers, and by C the complex plane. Given a space S, a point c ∈ S and a number e > 0, we denote by BS(c, e) the open ball in S with center c and radius e. 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., [5, (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 , and by F1(X) the hyperspace of singletons. The reader is referred to Nadler’s book [5] 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 ) defined by 2(A) = f(A) for every A ∈ 2 and C(f)(A) = f(A) for every A ∈ C(X). A mapping between continua is said to be: — open provided the image of an open subset of the domain is open in the range; — monotone provided the point-inverses are connected; — light provided the point-inverses are zero-dimensional. The following theorem is the main result of this paper. 1. Theorem. Let a continuum X be locally connected, and a mapping f : X → Y be such that the induced mapping C(f) : C(X) → C(Y ) is open. Then f is monotone. Received by the editors June 16, 1997 and, in revised form, January 4, 1998. 1991 Mathematics Subject Classification. Primary 54B20, 54F15, 54E40.

Openness and monotoneity of induced mappings