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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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A continuum is a connected compact metric space. The second symmetric product of a continuum X, F 2 (X), is the hyperspace of all nonempty subsets of X having at most two elements. An element A of F 2 (X) is said to make a hole with respect to multicoherence degree in F 2 (X) if the multicoherence degree of F 2 (X) − {A} is greater than the multicoherence degree of F 2 (X). In this paper, we characterize those elements A ∈ F 2 (X) such that A makes a hole with respect to multicoherence degree in F 2 (X) when X is a cyclicly connected graph.