Induced mappings on symmetric products, some answers

Given a continuum X, let Fn(X) denote the hyperspace of nonempty subsets of X with at most n points. For n≥2, let SFn(X)=Fn(X)/F1(X) be the quotient space. Given a mapping between continua f:X→Y, we consider the induced mappings fn:Fn(X)→Fn(Y) and Sfn:SFn(X)→SFn(Y). Given a class of mappings M, in this paper we consider relations between the statements f∈M, fn∈M and Sfn∈M, and we answer some questions about these relations considering the following classes of mappings: almost monotone, atriodic, freely decomposable and joining.