On a family of dendrites (Q5957250)

Let \(\mathcal D\) be a family of dendrites (i.e. locally connected continua containing no simple closed curve) and \(\mathbf{O}\) a class of open mappings. The relation \(\leq _{\mathbf{O}}\) defined by ``\(Y \leq _{\mathbf{O}} X\) if there exists an open surjection of \(X\) onto \(Y\)'' is a quasi-ordering on \(\mathcal D\). A dendrite \(X_0\) is said to be unique minimal with respect to \(\mathbf{O}\) if the following holds: for each dendrite \(Y\) if \(Y \leq _{\mathbf{O}} X_0\) then \(Y\) is homeomorphic to \(X_0\). NEWLINENEWLINENEWLINEThis paper is a continuation of the studies by \textit{J. J. Charatonik}, \textit{W. J. Charatonik} and \textit{J. R. Prajs} [Mapping hierarchy for dendrites, Diss. Math. 333 (1994; Zbl 0822.54009)]. The author constructs a countable family \(\mathcal F\) of dendrites and investigates the open images of members of it. The above construction is modelled on the one described in \textit{D. Opěla}, \textit{P. Pyrih} and \textit{R. Šámal} [Quest. Answers Gen. Topology 18, No.~2, 229-236 (2000; Zbl 0967.54035)]. NEWLINENEWLINENEWLINEIt is proved that among all members of \(\mathcal F\) there are only two minimal elements of the class \({\mathcal D}_{\leq _{\mathbf{O}}}\) and only one of them is unique minimal with respect to open mappings.