Induced mappings between quotient spaces of symmetric products of continua

Given a continuum X and n ∈ N . Let H ( X ) ∈ { 2 X , C ( X ) , F n ( X ) } be a hyperspace of X, where 2 X , C ( X ) and F n ( X ) are the hyperspaces of all nonempty closed subsets of X, all subcontinua of X and all nonempty subsets of X with at most n points, respectively, with the Hausdorff metric. For a mapping f : X → Y between continua, let H ( f ) : H ( X ) → H ( Y ) be the induced mapping by f, given by H ( f ) ( A ) = f ( A ) . On the other hand, for 1 ⩽ m < n , SF m n ( X ) denotes the quotient space F n ( X ) / F m ( X ) and similarly, let SF m n ( f ) denote the natural induced mapping between SF m n ( X ) and SF m n ( Y ) . In this paper we prove some relationships between the mappings f, 2 f , C ( f ) , F n ( f ) and SF m n ( f ) for the following classes of mapping: atomic, confluent, light, monotone, open, OM, weakly confluent, hereditarily weakly confluent.