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Abstract. For a continuum X the hyperspace of nonempty closed subsets of X with at most n components is called the n-fold hyperspace Cn(X) and if m < n then Cm(X) ⊂ Cn(X) so it is possible to form a quotient space Cn(X)/Cm(X) identifying the set Cm(X) to a point in Cn(X). If f is a mapping from a continuum X onto a continuum Y there will be a induced mappings between Cn(X) and Cm(X) and between the quotient spaces Cn(X)/Cm(X) and Cn(Y )/Cm(Y ). Now if a list of function properties that are of interest to continua theorists is considered, there will be natural questions about when these properties are passed on from the functions between the continua to the induced mappings between the hyperspaces or the induced mappings between the quotients of the hyperspaces. Many of these questions have been considered extensively for the hyperspaces so the main thing that is new here is the questions and answers about the quotient spaces and their induced mappings. Here we consider the following families of mappings:

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In this paper we discuss the notions of pseudo-contractibility and weak contractibility on hyperspaces of (Hausdorff) continua. Also we prove that if a continuum X contains an R i-set then it is not pseudocontractible. As a consequence we have that the existence of an R i-set in a continuum X implies non(pseudo)-contractibility of some hyperspaces.