Hyperspaces of arcs containing a point
Keywords:
Continuum, dendrite, fan, hyperspace of arcs and singletons, hyperspaces of arcs containing a point, smooth fanAbstract
For a metric continuum $X$ and for $p \in X$, we define the hyperspace $\operatorname{Arcs}(p, X)$ as the set of all arcs in $X$ that contain the point $p$, endowed with the Hausdorff metric. We show geometric models of $\operatorname{Arcs}(p, X)$, in the case when $X$ is the arc, the simple closed curve, the simple triod, the noose and the $\sin \left(\frac{1}{x}\right)$-continuum. Moreover, we present a characterization of dendrites in terms of this hyperspace.
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