The cone = hyperspace property, a characterization

Let X be a continuum. Let C(X) be the hyperspace of subcontinua of X. We say that X is said to have the cone = hyperspace property if there exists a homeomorphism h :C(X)→Cone(X) such that h(X)=vertex of (Cone(X)) and h({x})=(x,0) for each x∈X. In this paper we prove. m. Let X be a finite-dimensional continuum. Then the following are equivalent: has the cone = hyperspace property, and ere is a selection s :C(X)−{X}→X such that, for every Whitney level A for C(X), s|A :A→X is a homeomorphism. w some consequences of this theorem.