A continuum \(X\) such that \(C(X)\) is not continuously homogeneous. (Q2798073)

A continuum \(X\) is said to be \textit{continuously homogeneous} if for any two points \(p,q \in X\) there exists a surjective mapping \(f:X \rightarrow X\) such that \(f(p)=q\). For a given metric continuum \(X\) the symbols \(2^{X}\) and \(C(X)\) denote the hyperspace of nonempty closed subsets of \(X\) and the hyperspace of nonempty subcontinua of \(X\), respectively, endowed with the Hausdorff metric.NEWLINENEWLINE In [Bull. Pol. Acad. Sci., Math. 32, 339--342 (1984; Zbl 0561.54026)] \textit{W. J. Charatonik} and \textit{Z. Garncarek} proved that for every continuum \(X\) the hyperspace \(2^{X}\) is continuously homogeneous and asked if \(C(X)\) is continuously homogeneous as well.NEWLINENEWLINE In the reviewed paper the author answers this question in the negative by showing a continuum as in the title.