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

TL;DR

This paper characterizes when the hyperspace of subcontinua containing a point in a continuum is unique up to homeomorphism, showing it occurs precisely when the continuum is a tree within certain classes.

Contribution

It provides a complete characterization of continua with unique hyperspaces relative to dendrites, answering open questions from prior research.

Findings

Unique hyperspace occurs only for trees among dendrites.

Some classes of continua do not have unique hyperspaces.

The results clarify the structure of hyperspaces in continuum theory.

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 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 \cite{Corona.et.al(2019)}.