About the uniqueness of the hyperspaces $C(p,X)$ in some classes of continua

Abstract

Given a continuum X and p ∈ X, we will consider the hyperspace C(p, X) of all subcontinua of X containing p. Given a family of continua C, a continuum X ∈ C and p ∈ X, we say that (X, p) has unique hyperspace C(p, X) relative to C if for each Y ∈ C and q ∈ Y such that C(p, X) and C(q, Y ) are homeomorphic, then there is an homeomorphism between X and Y sending p to q. In this paper we show that (X, p) has unique hyperspace C(p, X) relative to the classes of dendrites if and only if X is a tree, we present also some classes of continua without unique hyperspace C(p, X); this answer some questions posed in [2].