Induced mappings between hyperspaces \(HS (p, X)\) of continua (Q776391)

The authors consider the following notion. Let \(X\) be a continuum and \(p\in X\). By \(C(X)\) one means the hyperspace of subcontinua of \(X\), \(C(p,X)\) the subspace of elements of \(C(X)\) that contain \(p\), and \(HS(p,X)\) the quotient space \(C(X)/C(p,X)\). There is a quotient map \(\pi_p^X:C(X)\to C(p,X)\). If \(f:X\to Y\) is a map between continua, then \(C(f):C(X)\to C(Y)\) is the map given by \(C(f)(A)=f(A)\). It induces a map \(H(p,f):HS(p,X)\to HS(f(p),Y)\), and one has that \(\pi_{f(p)}^Y\circ C(f)=HS(p,f)\circ\pi_p^X\). The interest of the paper involves consideration of a class \(\mathcal{M}\) of maps, and then working on the problem of when any of the following statements implies the other: \begin{enumerate}\item\(f\in\mathcal{M}\). \item \(C(f)\in\mathcal{M}\). \item \(HS(p,f)\in\mathcal{M}\) for some \(p\in X\). \end{enumerate} Among the classes of maps studied are: monotone, confluent, weakly confluent, open, light, and atomic. Section 3 is devoted to examples, and Section 4 deals with the case that \(\mathcal{M}\) consists of injections, surjections, and bijections which in this case are homeomorphisms. Sections 5--9 engage, respectively, monotone maps, confluent and weakly confluent maps, open maps, light maps, and atomic maps.