Abstract
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. Theorem. Let X be a finite-dimensional continuum. Then the following are equivalent : (a) X has the cone = hyperspace property, and (b) there 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. We show some consequences of this theorem.
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