Concerning the cone = hyperspace property

An example of a one-dimensional, nonchainable, noncircle-like continuum is constructed and shown to have the cone = hyperspace property. In addition, it is proved that a sufficient condition for a continuum X to have the cone = hyperspace property is that there exists a selection for C(X){X} which, for some Whitney map for C(X) , maps each nondegenerate Whitney level homeomorphically onto X . The class of Whitney stable continua includes the arc, the circle, the pseudo-arc, and the solenoids. It is shown that a certain example due to W. T. Ingram of a nonchainable, atriodic, tree-like continuum is also Whitney stable.