Irreducibility of strong size levels (Q6550120)

For a metric continuum \(X\) let \(F_{n}(X)\) denote the hyperspace of nonempty closed subsets of \(X\) with at most \(n\) points, and let \(C_{n} (X)\) denote the hyperspace of nonempty closed subsets of \(X\) with at most \(n\) components. These hyperspaces are topologized with the Hausdorff metric. \par A \textit{Whitney map} is a continuous function \(\mu : C_{n}(X) \to [0, 1]\) such that: (a) for each \(p \in X\), \(\mu (p) = 0\); (b) \(\mu (X) = 1\); and (c) if \(A,B \in C_{n}(X)\) and \(A \subsetneq B\), then \(\mu (A) < \mu (B)\). Sets of the form \(\mu ^{-1} (t)\) where \(t \in [0,1]\) are \textit{Whitney levels}. \newline A \textit{strong size map} is a continuous function \(\sigma : C_{n}(X) \to [0, 1]\) such that: (a) for each \(A \in F_{n}(X)\), \(\sigma (A) = 0\); (b) \(\sigma (X) = 1\); and (c) if \(A,B \in C_{n}(X)\), \(A \subsetneq B\) and \(B \notin F_{n}\), then \(\mu (A) < \mu (B)\). Sets of the form \(\sigma ^{-1} (t)\) where \(t \in [0,1)\) are \textit{strong size levels}. \N\NIn [An. Inst. Mat., Univ. Nac. Auton. Mex. 26, 59--64 (1986; Zbl 0624.54007)] \textit{A. Illanes} showed that if \(X\) is a continuum having a Whitney level irreducible with respect to a finite subset, then \(X\) is also irreducible with respect to a finite set. In [Topology Appl. 160, 1816--1828 (2013; Zbl 1285.54004)], \textit{L. Peredes-Rivas} and \textit{P. Pellicer-Covarrubias} showed an irreducible, hereditarily decomposable continuum such that none of its proper strong size levels is irreducible. In [ibid. 283, Article ID 107339, 12 p. (2020; Zbl 1467.54007)], \textit{F. Capulín-Perez} et al. asked whether there exists a continuum with irreducible strong size levels. In the paper under review the authors give the answer to this question by proving the following theorem: Let \(X\) be a continuum and \(n \geqslant 2\). Then the strong size levels for \(C_{n}(X)\) are not irreducible.