LAUSR

LetX be a topological space and CL(X) be the family of all nonempty closed subsets of X . In this paper we discuss the problem of when a continuous map between topological spaces induces a continuous function between their respective hyperspaces. As a main result we characterize the continuity of the induced function in the case of the Fell and Attouch-Wets hyperspaces. Additionally we explore the problem of whether a continuous action of a topological group G on a topological space X induces a continuous action on CL(X). In particular we give sufficient conditions on the topology of G to guarantee that the induced action on CL(X) is continuous, provided that CL(X) is equipped with the Hausdorff or the Attouch-Wets metric topology.