PROCEEDINCiS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 98. Number 3. November 1986 MONOTONE AND OPEN WHITNEY MAPS ALEJANDRO ILLANES ABSTRACT. In this paper we prove that if X is a locally connected continuum, then there exists a monotone and open Whitney map for 2X. 1. Introduction. A continuum is a nonempty compact connected metric space. A Peano continuum is a locally connected continuum. The hyperspaces of a con- tinuum X are the spaces 2X = {A C X: A is nonempty and compact} and C(X) = {Ag2x : A is connected} metrized with the Hausdorff metric. A Whitney map for the hyperspace )/ of X is a continuous function u> : M —► [0, oo) such that w({x}) = 0 for each x G X, and if A, B G M and A C B ^ A, then ui(A) < oj(B) is monotone if w-1(£) is connected for every t G [0, oo). It is known that every Whitney map for C(X) is monotone and open [3, p. 1032]. In [6] it was shown that even when X is a circle or an arc, Whitney maps for 2X can be constructed which are neither monotone nor open. S. B. Nadler, Jr. has asked in [6, 14.63 and 14.64] whether for every continuum X there exists a monotone (or an open) Whitney map for 2X. Recently, W. J. Charatonik [2] answered both questions negatively by showing an example of a continuum X in the plane which has neither monotone nor open Whitney maps for 2X. In this paper, we prove that if X is a Peano continuum, then a monotone open Whitney map can be constructed for 2X. Throughout this paper X will denote a continuum with a metric d. A metric p for X is said to be convex provided that, given x,y G X, there exists z G X such that p(x,z) = p(x,y)/2 = p(y,z). It is easy to prove that if p is a convex metric for X, then, for any x,y G X, there exists a isometry a: [0,p(x,y)] —► X such that cr(0) = x and a(p(x,y)) = y. X is said to admit a convex metric p provided that p is a convex metric for X and the (original) topology on X is the same as the topology for X obtained by p. If X admits a convex metric p, then X is a Peano continuum [4]. The converse was proved independently by R. H. Bing [1] and E. E. Moise [5]. 2. A monotone open Whitney map. Given a positive integer n and A G 2X, we define w„(A) = inf{e > 0: there exist Xi,..., xra G X such that A C Be(xi) U ••• U B£(xn)}, where B£(x) = {y G X: d(x,y) < e}. Then un: 2X -> [0,oo) is continuous, wn({x}) = 0 for each x G X and wn(A) < diameter of X. So, if we define u>: 2X —► [0, oo) by uj(A) = £)u;n(A)/2™, then w is continuous and oj({x}) —0 for each x G X. To prove that w is a Whitney map, we take A, B G 2X such that A c B ^ A. Let 6 be a point of B - A and let e > 0 such that B2e(b) n A = 0. Let m be Received by the editors July 10, 1985 and, in revised form, October 2, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 54B20, 54F25. Key words and phrases. Continuum, hyperspace, Whitney map. ©1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page 516 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use