Whitney stability and contractible hyperspaces

Abstract

A contraction technique of the first author is used to derive, for a certain class of continua with contractible hyperspaces, results about Whitney stability, the cone = hyperspace property, and Whitney properties. Let C(X) be the hyperspace of subcontinua of the continuum X, and let ,u be a Whitney map on C(X). Kelley [61 has shown that if C(X) is contractible, then the contraction can be chosen so that each subcontinuum of X grows until it becomes X. The first author [3, 4] has improved this result by showing that the contraction may be chosen with the additional property that, at time t, the contraction has moved each point A of C(X) with ,u(A) t. In this paper, we exploit this contraction in several ways. We prove that if the continuum X has property K, has the covering property, and has arc components that are one-to-one continuous images of the real line such that both ends are dense in X, then the cone over X, Cone(X), is homeomorphic to C(X). Moreover, if pL is a Whitney map on C(X), then the homeomorphism may be chosen so that its restriction to X x { t} is a homeomorphism onto tcl(t). In particular, the base of the cone is mapped homeomorphically onto the singletons and the vertex onto the point X. Hence, such continua have the cone = hyperspace property and are Whitney stable. We give an example of a continuum with these properties that is neither arc-like nor circle-like. This gives a negative answer to Question 8.14 of [9]. The solenoids are examples of circle-like continua with these properties. We turn our attention to the more general class of continua with property K and with the property that each proper, nondegenerate subcontinuum is an arc. We construct a monotone map of Cone(X) onto C(X) similar to the homeomorphism above. In the case that X is the inverse limit of arcs with open maps as the bonding maps, we approximate this monotone map with a level-preserving homeomorphism. It follows that such a continuum has the cone = hyperspace property and is Whitney stable. This gives an affirmative answer to Question 14.22 of [9] and partial answers to Questions 8.11 and 8.15 of [9]. Modifications of these methods are used to show that the sin 1/x-curve and a ray spiralling on a circle do not admit essentially different Whitney maps. Received by the editors December 12, 1980 and, in revised form, February 23, 1981. 1980 Mathematics Subject Classification. Primary 54B20.