Induced mappings on hyperspaces \(C(p,X)\) and \(K(X)\)
For a metric continuum \(X\) and a point \(p \in X\), consider the following hyperspaces: \(C(X)=\{A \subset X : A\) is a subcontinuum of \(X\}\), \(C_{p} (X) = \{A \in C(X) : p \in A\}\), and \(K(X)=\{C_{p} (X) \in C(C(X)) : p \in X\}\). Given a mapping between continua \(f:X \to Y\), the authors consider the following natural mappings: -- \(\overline{f} : C(X) \rightarrow C(Y)\), given by \(\overline{f} (A)=f(A)\) (the image of \(A\) under \(f\)), -- \(\tilde{f} : K(X) \rightarrow K(Y)\), given by \(\tilde{f} (C(p,X)) = C(f(p),Y)\), -- \(\Check{f} : K(X) \rightarrow C(C(Y))\) given by \(\Check{f} (C(p,X))=\overline{f}(C(p,X))\), and -- \(\overline{f}_{p} : C(p,X) \rightarrow C(f(p),Y)\) given by \(\overline{f}_{p}(A)=\overline{f}(A)\). Now let \(\mathcal{M}\) denote a class of mappings between continua. The main problem studied in this paper is to find all interrelations betweeen the following statements: (0) \(f \in \mathcal{M}\), (1) \(\overline{f} \in \mathcal{M}\), (2) \(\tilde{f} \in \mathcal{M}\), (3) \(\Check{f} \in \mathcal{M}\), and (4) \(\overline{f}_{p} \in \mathcal{M}\) for all \(p \in X\). There are a number of papers and authors that have studied the respective problem for many possible hyperspaces. The authors include a very complete list of references. In the paper under review the authors study this problem for the classes of monotone, confluent, weakly confluent, light and open mappings. They include many results on this topic and pose several interesting questions.
scientific article; zbMATH DE number 6804707
scientific article