Induced maps on \(n\)-fold symmetric product suspensions
Given a metric continuum \(X\), the \(n\)-symmetric product \(F_n(X)\) is defined as the hyperspace of nonempty subsets of \(X\) with at most \(n\) elements. The author considers the so called \(n\)-fold symmetric suspension \(SF_n(X)\) of the continuum \(X\) defined as the quotient space \(F_n(X)/F_1(X)\). Given a continuous function between continua \(f : X\to Y\), one can consider the mapping \(f_n : F_n(X)\to F_n(Y)\) given by \(f_n(A)=f(A)\) (the image of \(A\) under \(f\)). This mapping induces, in the natural way, a mapping \(Sf_n : SF_n(X)\rightarrow SF_n(Y)\). Given a class \(\mathcal M\) of mappings, the author makes a detailed study of the relationships among the statements (1) \(f\in \mathcal M\), (2) \(f_n\in \mathcal M\), and (3) \(Sf_n\in \mathcal M\), for several classes of mappings as: monotone, open, confluent, light, etc. Many of the results in this paper concerning the relationships between the statements (1) and (2) have also recently appeared in [\textit{G. Higuera} and \textit{A. Illanes}, Topol. Proc. 37, 367--401 (2011; Zbl 1282.54009)]. The paper includes several interesting questions.
scientific article; zbMATH DE number 5918209
scientific article