Monotone classes of dendrites (Q2810699)

A map \(f : X \rightarrow Y\) from a continuum \(X\) into a continuum \(Y\) is said to be \textit{monotone} if \(f^{-1}(y)\) is connected for all \(y \in f(X)\). Two continua \(X\) and \(Y\) are said to be \textit{monotone equivalent} if there exist monotone maps from \(X\) onto \(Y\) and from \(Y\) onto \(X\). A continuum \(X\) is said to be \textit{isolated with respect to monotone maps} if for each continuum \(Y\) the existence of two monotone mappings, one from \(X\) onto \(Y\) and the other from \(Y\) onto \(X\) implies that \(X\) and \(Y\) are homeomorphic. In this paper the authors prove that a dendrite \(D\) is isolated with respect to monotone maps if and only if the set of ramification points of \(D\) is finite. This is the answer to the Problem 6.1 of [\textit{J. J. Charatonik}, Topology Appl. 38, No. 2, 163--187 (1991; Zbl 0726.54012)].NEWLINENEWLINEA relation is a \textit{quasi-order} if it is reflexive and transitive. A relation is a \textit{partial order} if it is reflexive, antisymmetric and transitive. Let \(\leq \) be a relation defined as follows: \(D_{1} \leq D_{2}\) if and only if there exists a monotone onto map \(f : D_{2} \rightarrow D_{1}\). The family of all dendrites is quasi-ordered by the relation \(\leq \). The family of dendrites with a finite number of ramification points is, as the authors show, partially ordered by the relation \(\leq\).