Uniqueness of the hyperspaces $C(p,X)$ in the class of trees

Abstract

Given a continuum $X$ and $p\in X$, we will consider the hyperspace $C(p,X)$ of all subcontinua of $X$ containing $p$. Given a family of continua $\mathcal{C}$, a continuum $X\in\mathcal{C}$ and $p\in X$, we say that $(X,p)$ has unique hyperspace $C(p,X)$ relative to $\mathcal{C}$ if for each $Y\in\mathcal{C}$ and $q\in Y$ 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 study some topological and geometric properties about the structure of $C(p,X)$ when $X$ is a tree, being the main result that $(X,p)$ has unique hyperspace $C(p,X)$ relative to the class of trees.