An example of openly minimal dendrite (Q2703098)

A continuum is a nonempty compact connected metric space. A simple closed curve is any space which is homeomorphic to the unit circle. A dendrite is a locally connected continuum containing no simple closed curve. An arc is any space which is homeomorphic to \([0,1].\) An arc with endpoints \(p\) and \(q\) is denoted \(pq\). An arc \(pq\) in a continuum is an antenna attached to \(p\) provided \(pq\smallsetminus\{p\}\) is open. A continuous surjection \(f:X\to Y\) is open provided that for each open subset \(U\) of \(X\), \(f(U)\) is open in \(Y\). A dendrite \(X\) is openly minimal provided that every nondegenerate open image of \(X\) can be openly mapped onto \(X\). For \(r\in\{3,4,\dots,\omega\}\) and \(t\in\{0,1,2,\dots,\omega\}\), the authors define a collection of Country dendrites \(C^t_r\) with ramification points or \(r\) connections and \(t\) antennas. The authors show that the Country dendrite \(C^1_\omega\) is openly minimal and is not homeomorphic to all its nondegenerate open images, answering a question of \textit{J. J. Charatonik}, \textit{W. J. Charatonik} and \textit{J. R. Prajs} [Mapping hierarchy for dendrites, Diss. Math. 333 (1994; Zbl 0822.54009)] in the negative.