A complete classification of fruit trees sharing the same hyperspace C(p, X)

Year 2025, Early Access, 1 - 18

Javier Sánchez-martínez , Florencio Corona-vázquez , Russell-aarón Quiñones-estrella

https://doi.org/10.15672/hujms.1422352

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, there is a 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 fruit tree; we show that, in this class of trees, there is never uniqueness of the corresponding hyperspaces, being the noose and the arc the only two exceptions. We provide also the construction of all fruit trees $G$ and $q\in G$ such that $C(p,X)$ is homeomorphic to $C(q,G)$.

Keywords

Continua , hyperspaces , finite graphs , fruit trees , unique hyperspaces

Supporting Institution

Universidad Autónoma de Chiapas

References

• [1] F. Corona–Vázquez, R. A. Quiñones–Estrella and J. Sánchez–Martínez, About the uniqueness of the hyperspaces C(p, X) in some classes of continua, Topology Appl., 322, 108313, 2022.