Dendrites which are determined by their positive Whitney levels (Q1616991)

Given a continuum \(X\), we consider the set \(C(X)\) formed by the subcontinua of \(X\) endowed with the Hausdorff metric. A Whitney map for \(C(X)\) is a map \(\mu: C(X)\to [0,1]\) satisfying the following properties: (1) \(\mu(A)=0\) if and only if \(A\) is degenerate. (2) \(\mu(A)<\mu(B)\), for any two \(A,B\in C(X)\) such that \(A\subset B\) and \(A\neq B\). (3) \(\mu(X)=1\). If \(\mu\) is a Whitney map for \(C(X)\) and \(t\in [0,1]\), then \(\mu^{-1}(t)\) is called a Whitney level. A positive Whitney level of \(C(X)\) is a set of the form \(\mu^{-1}(t)\), where \(\mu\) is a Whitney map and \(t \in (0, \mu(X))\). Let \(\mathcal{WL}(X)\) be the set of positive Whitney levels of \(C(X)\). We say that a continuum \( X\) is Whitney equivalent to a continuum \(Y\) if the sets \(\mathcal{WL}(X)\) and \(\mathcal{WL}(Y)\) are topologically equal. Moreover, we say that a continuum \(X\) is determined by its Whitney levels (or, for short, Whitney determined) provided that, for each continuum \(Y\), the following holds: if \(Y\) is Whitney equivalent to \(X\), then \(X\) and \(Y\) are homeomorphic. In [Topol. Proc. 39, 293--315 (2012; Zbl 1271.54051)], \textit{A. Illanes} and \textit{R. Leonel} asked the following: Are dendrites with closed set of end points Whitney determined? The purpose of the paper under review is to answer that question in the positive.