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

Abstract

Let $X$ be a continuum and $p \in X$. We consider the hyperspace $C(p,X)$ consisting of all subcontinua of $X$ that contain the point $p$. Given a family of continua $\mathcal{C}$, a continuum $X \in \mathcal{C}$, and a point $p \in X$, we say that $(X, p)$ has a unique hyperspace $C(p, X)$ relative to $\mathcal{C}$ if, for any $Y \in \mathcal{C}$ and $q \in Y$ such that $C(p,X)$ and $C(q,Y)$ are homeomorphic, there exists a homeomorphism from $X$ to $Y$ sending $p$ to $q$. In this paper, we investigate the topological and geometric structure of $C(p,X)$ when $X$ is a fruit tree. We show that, except for the arc and the noose, no fruit tree has a unique hyperspace in this class. Furthermore, we construct all fruit trees $G$ and points $q \in G$ 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

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