On the unicoherence of \(F_n(X)\) and \(SF^n_m(X)\) of continua (Q2850658)

For a metric continuum \(X\), let \(F_{n}(X)\) be the hyperspace of all nonempty subsets of \(X\) with at most \(n\) elements, if \(0<m<n\) let \(SF_{m}^{n}(X)=F_{n}(X)/F_{m}(X)\), that is, \(SF_{m}^{n}(X)\) is the space obtained by shrinking to a point the subset \(F_{m}(X)\) of \(F_{n}(X)\). In this paper, the authors continue the study of unicoherence of continua of the form \(F_{n}(X)\) and \(SF_{m}^{n}(X)\). Answering a question by J. J. Charatonik, posed in Problem 3 of [\textit{E. Castañeda}, Topol. Proc. 23 (Spring), 61--67 (1998; Zbl 0987.54043)], they show that if \(X\) is a \(\lambda\)-dendroid, then \(F_{2}(X)\) is unicoherent. They also show that if \(2<n\), then \(SF_{m}^{n}(X)\) is unicoherent and they give conditions under which \(SF_{1}^{2}(X)\) is unicoherent. In [\textit{F. Barragan}, Topology Appl. 158, No. 10, 1192--1205 (2011; Zbl 1223.54012)] a continuum \(Y\) was given such that \(SF_{1}^{2}(Y)\) is not unicoherent. In the paper under review, the authors show that Barragan's example is wrong by proving that \(SF_{1}^{2}(Y)\) is unicoherent. They also show that if \(Z\) is the \(\theta\)-curve, then \(SF_{1}^{2}(Z)\) is unicoherent. This example is also wrong as it has very recently been proved by the first named author and the reviewer (in a paper to appear), who in fact have shown that, for each continuum \(X\), \(SF_{1}^{2}(X)\) is unicoherent.