Some relationships between induced mappings (Q989084)

For a continuum \(X\) and for a positive integer \(n\) let \(2^X\) denote the hyperspace of nonempty closed subsets; let \(C_{n}(X)\) denote the hyperspace of all nonempty closed subsets of \(X\) with at most \(n\) components, and \({\mathcal F}_{n}(X)\) the hyperspace of nonempty finite sets with at most \(n\) points. The \textit{\(n\)-fold hyperspace suspension} \(HS_{n}(X)\) is the quotient space \(C_{n}(X)/{\mathcal F}_{n}(X)\). For a given map \(f:X \rightarrow Y\) between continua \(X\) and \(Y\), let \(2^f:2^X\rightarrow 2^Y\), \(C_{n}(f):C_{n}(X) \rightarrow C_{n}(Y)\) and \(HS_{n}(f):HS_{n}(X) \rightarrow HS_{n}(Y)\) denote the corresponding induced maps. In the paper under review, the author continues the studies made by \textit{J. J. Charatonik, A. Illanes} and \textit{S. Macias} [Houston J. Math. 33, 1047--1057 (2007; Zbl 1021.54011)]; \textit{S. Macias} [Topol. Proc. 28, 143--152 (2004; Zbl 1081.54010)]; \textit{M. de J. López} and \textit{S. Macias} [Houston J. Math. 33, 1047--1057 (2007; Zbl 1158.54002)]. He finds relationships between \(f\), \(C_{n}(f)\), \(HS_{n}(f)\) and \(2^f\) when he considers the following classes of mappings: monotone, OM, confluent, semi-confluent, weakly confluent, pseudo-confluent, quasi-monotone, weakly monotone or joining.