PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 110, Number 4, December 1990 ARC-SMOOTHNESS AND CONTRACTIBILITY IN WHITNEY LEVELS ALEJANDRO ILLANES (Communicated by James E. West) Abstract. Let X be a continuum. Let 2X (resp., C(X)) be the space of all nonempty closed subsets (resp., subcontinua) of X . In this paper we prove that if X is an arc-smooth continuum, then there exists an admissible Whitney map p: 2X —> R such that ¡i\C{X): C{X) —» R is admissible and for every t e (0, p{X)), p~X{t) and (p\C(X))~l(t) are arc-smooth. This answers a question by J. T. Goodykoontz, Jr. Also we give an example of a contractible continuum X such that, for every Whitney map v. C(X) —» R there exists te(0,u{X)) suchthat u~l(t) is not contractible. 0. Introduction A continuum is a nondegenerate compact connected metric space. The hy- perspaces of a continuum X are the spaces 2X = {A c X: A is nonempty and closed in X} and C(X) — {A G 2X : A is connected} metrized with the Hausdorff metric H. We identify {{x}: x G X} c C(X) c 2X with X. A Whitney map for a hyperspace St of X is a continuous function p : A%A —> R (R is the real line) such that p({x}) = 0 for each x G X and if A c B ^ A , then p(A) < p(B). If p is a Whitney map for C(X) and 0 < t < p(X), then p~](t) is called a Whitney level; if 0 < t < p(X), then ¿i- (/) is called a positive Whitney level. A topological property P is called a Whitney property provided that whenever a continuum X has property P, so does p~l(t) for each Whitney map for C(X) and 0 < t < p(X). A Whitney map p is called an admissible Whitney map for %A [4] provided that there exists a continuous homotopy F: A? x [0, 1] -» %A satisfying: (a) For all A € &, F(A, 1) = A and F(A, 0)gX . (b) If p(F (A, t)) >0 for some A g J?* and r e[0, l],then ¿¿(F(,4,s)) < p(F(A, ?)) whenever 0 < 5 < /. Let /? e A. Then X is arc-smooth at p ([1] and [2]) provided there exists a continuous function a: X —> C(X) satisfying: (a) a(p) = {p} . Received by the editors July 28, 1988 and, in revised form, August 11, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 54B20. ©1990 American Mathematical Society 0002-9939/90 $1.00+ $.25 per page 1069 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use