On the continuity of the Hutchinson operator

Abstract

We investigate when the Hutchinson operator associated with an iterated function system is continuous. The continuity with respect to both the Hausdorffmetric and Vietoris topology is carefully considered. An example showing that the Hutchinson operator on the hyperspace of nonempty closed bounded sets need not be Hausdorffcontinuous is given. Infinite systems are also discussed. The work clarifies and generalizes several partial results scattered across the literature.