Induced mappings on the hyperspace of convergent sequences

The symbol Sc(X) denotes the hyperspace of all nontrivial convergent sequences in a Hausdorff space X. This hyperspace is endowed with the Vietoris topology. For a given mapping between Hausdorff spaces f:X→Y, define the induced mapping Sc(f) by Sc(f)(A)=f[A] (the image of A under f) for every A∈Sc(X). In the current paper, we study under which conditions the fact that f belongs to a given class of mappings M implies that Sc(f) belongs to M, and vice versa.