Hyperspaces of arcs containing a point (Q6546622)

A \textit{continuum} is a compact connected metric space. A \textit{subcontinuum} refers to a compact connected subspace of a continuum. The hyperspace of subcontinua of a continuum \(X\), denoted by \(C(X)\), is the space that consists of all subcontinua of \(X\) equipped with the Hausdorff metric.\N\NAn \textit{arc} is any topological copy of the closed interval \([0,1]\).\N\NLet \(X\) be a continuum and let \(p \in X\). The symbol \(Arcs(p,X)\) represents the subspace \(\{\{p\}\} \cup \{ A \in C(X) : A \ \text{is an arc and} \ p \in A\}\) of \(C(X)\). The purpose of this paper is to investigate the topological properties of \(Arcs(p,X)\). Among several results, the authors present geometric models of \(Arcs(p,X)\) when \(X\) is one the following spaces: the arc, the simple closed curve, the simple triod, the noose, and the \(\sin \frac 1 x\)-continuum. A consequence of the main results is the characterization of locally connected continua. Specifically, for a locally connected continuum \(X\), the following conditions are equivalent: (a) \(X\) is a dendrite, (b) \(Arcs(p,X)\) is compact of each \(p \in X\), and (c) there exists \(q \in X\) such that \(Arcs(q,X)\) is homeomorphic to \(X\).