# Hyperspaces $C(p,X)$ of finite graphs

## TL;DR

This paper explores the topological properties of hyperspaces of subcontinua in finite graphs, establishing conditions for homeomorphism, analyzing homogeneity, and constructing examples with specific characteristics.

## Contribution

It provides new conditions for when hyperspaces are homeomorphic, relates homogeneity to the size of hyperspaces, and constructs finite graphs with prescribed hyperspace properties.

## Key findings

- Conditions for homeomorphism of hyperspaces in finite graphs
- Relationship between graph homogeneity and hyperspace diversity
- Construction of finite graphs with specified hyperspace sizes

## Abstract

Given a continuum $X$ and $p\in X$, we will consider the hyperspace $C(p,X)$ of all subcontinua of $X$ containing $p$ and the family $K(X)$ of all hyperspaces $C(q,X)$, where $q\in X$. In this paper we give some conditions on the points $p,q\in X$ to guarantee that $C(p,X)$ and $C(q,X)$ are homeomorphic, for finite graphs $X$. Also, we study the relationship between the homogeneity degree of a finite graph $X$ and the number of topologically distinct spaces in $K(X)$, called the size of $K(X)$. In addition, we construct for each positive integer $n$, a finite graph $X_n$ such that $K(X_n)$ has size $n$, and we present a theorem that allows to construct finite graphs $X$ with a degree of homogeneity different from the size of the family $K(X)$.