Whitney preserving maps on finite graphs
For a metric continuum \(X\), let \(C(X)\) denote the hyperspace of subcontinua of \(X\). For a map \(f: X \rightarrow Y\) from a continuum \(X\) to a continuum \(Y\), let \(\hat{f}\) denote the induced map (from \(C(X)\) to \(C(Y)\) given by \(\hat{f}(A) = f(A)\) for all \(A \in C(X)\)). The map \(f: X \rightarrow Y\) is called Whitney preserving if there exist two Whitney maps \(\mu : C(X) \rightarrow \mathbb{R} \) and \(\nu : C(Y) \rightarrow \mathbb{R} \) such that for every \(t \in [0, \mu(X)]\) there exists \(s \in [0, \nu(Y)]\) such that \(\hat{f}(\mu^{-1}(t))=\nu^{-1}(s)\). The first author proved in [Topology Appl. 126, 351--358 (2002; Zbl 1019.54016)] that a Whitney preserving map from an arcwise connected continuum onto the unit interval is a homeomorphism. In this paper the result is extend as follows: 1) a Whitney preserving map from an arcwise connected continuum onto a finite graph, which is not a simple closed curve, is a homeomorphism; 2) if \(f: X \rightarrow S^{1}\) is a Whitney preserving map from a locally connected continuum \(X\) onto a simple closed curve \(S^{1}\), then \(X\) is homeomorphic to the unit interval or \(X\) is homeomorphic to \(S^{1}\). The authors also show that every hereditarily weakly confluent Whitney preserving map between locally connected continua is a homeomorphism.
scientific article; zbMATH DE number 5888206
scientific article