Models for \(C_{p}(x)\) for atriodic continua
A continuum is a compact connected metric space with more than two elements. Given a continuum \(X\), let \(C(X)\) be the hyperspace of subcontinua of \(X\), considered with the Hausdorff metric. Given a point \(p\in X\), let \(C_p(X)=\{ A\in C(X):p\in A\}\). A subcontinuum \(A\) of \(X\) is called a triod provided that there exists a subcontinuum \(B\) of \(A\) such that \(A-B\) has at least three components. The continuum \(X\) is atriodic provided that it contains no triods. Given a subset \(W\) of a hyperspace, it is usually useful to find a model for \(W\) (i.e., a topological space homeomorphic to \(W\) described in a simpler way). In this direction, \textit{P. Pellicer-Covarrubias} [Houston J. Math. 31, 403--426 (2005; Zbl 1085.54008)] proved that if \(X\) is an atriodic continuum and \(p\in X\), then there exists a continuous function \(g: [0,1]\to [0,1]\) such that \(g(0)=0=g(1)\) and \(C_p(X)\) is homeomorphic to the set \(L(g)=\{(r,s)\in [0,1]^2 : 0\leq s\leq g(r)\}\). She also asked if the converse holds. In the paper under review, the author answers Pellicer-Covarrubias' question in the positive by constructing, for each \(g\), an atriodic (in fact, chainable) continuum \(X\) and a point \(p\in X\) such that \(L(g)\) is homeomorphic to \(C_p(X)\).
scientific article; zbMATH DE number 5080158
scientific article