Hyperspaces of arcs and two-point sets in dendroids (Q5957350)

A dendroid is an arcwise connected hereditarily unicoherent continuum. A dendrite is a locally connected dendroid. A dendroid with exactly one ramification point is called a fan. NEWLINENEWLINENEWLINELet \(X\) be a dendroid. Let \({\mathcal A}(X)=\{A \subset X : A\) is an arc or a one-point set\(\}\) and \(F_2 (X) = \{A \subset X : A\) is nonempty and has at most two points\(\}\). We have a natural bijection \(g: F_2 (X) \rightarrow {\mathcal A}(X)\) given by \(g(\{p\})=\{p\}\) and \(g(\{p,q\})=pq\), where \(pq\) is the unique arc with end points \(p\) and \(q\). For any dendrite \(X\) the function \(g\) is continuous and thus \(F_2 (X)\) and \({\mathcal A}(X)\) are homeomorphic. There is a question if the dendrites are the only dendroids \(X\) for which \(F_2 (X)\) and \({\mathcal A}(X)\) are homeomorphic. It is shown in this paper that for fans the question has a positive answer. Namely, it is proved that if \(X\) is a fan such that \(F_2 (X)\) and \({\mathcal A}(X)\) are homeomorphic, then \(X\) is a dendrite.