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

## TL;DR

This paper investigates the structure of hyperspaces of subcontinua containing a point in trees and proves that such hyperspaces are uniquely determined by the tree and the point, within the class of trees.

## Contribution

It establishes that for trees, the hyperspace $C(p,X)$ uniquely characterizes the pair $(X,p)$ among all trees, revealing a rigidity property.

## Key findings

- Hyperspaces $C(p,X)$ are uniquely determined by the tree and point.
- The main result confirms the uniqueness of $C(p,X)$ within the class of trees.
- Provides topological and geometric insights into hyperspaces 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.