Finitely generated Whitney mappings (Q2664080)

For a metric continuum \(X\) let \(C (X)\) denote the hyperspace of subcontinua of \(X\) with the Hausdorff metric. A Whitney mapping is a continuous function \(\mu : C(X) \to [0, \infty)\) such that: (a) for each \(p \in X\), \(\mu (p) = 0\) and, (b) if \(A,B \in C(X)\) and \(A \subset B\), then \(\mu (A) < \mu (B)\). This paper is motivated by the question: For which continua \(X\) is it possible to define a simple Whitney mapping for \(C(X)\)? (For instance, the diameter map \(diam : C([0,1]) \to [0,1]\) is a simple Whitney map.) The authors study the continua \(X\) for which there exist finitely generated Whitney mappings. The Whitney mapping \(\mu\) is \textit{finitely generated} if there exists a finite number of continuous functions \(f_{1}, \ldots , f_{n} : X \to [0,1]\) such that for each \(A \in C(X)\), \(\mu (A) = \mathrm{length}(f_{1}(A)) + \cdots + \mathrm{length}(f_{n}(A))\). For a tree \(X\), the authors find relations among the number of necessary mappings to generate a Whitney mapping with: the number of necessary arcs for covering \(X\); the number of end-points of \(X\); the disconnection number of \(X\); the dimension of \(C(X)\) and the number \(n\) for which \(X\) is an \(n\)-od. This is a partial answer to the Problem 2 of the paper [\textit{A. Illanes}, ``Hyperspaces of continua'', in: \textit{E. Pearl} (ed.), Open problems in topology. II. Amsterdam: Elsevier (2007; Zbl 1158.54300)].