Summary: Let , be Hausdorff topological spaces and , respectively the families of all non empty closed subsets of and with some hypertopologies assigned. For each function there is a natural function: , where for each , . In this paper, we study the relationship between and . We are primarily interested in finding necessary and sufficient conditions on to ensure the continuity of . To avoid trivial situations, we will assume that contains at least two points and an arc and is a surjection. Since the base spaces are embedded in their hyperspaces, we always have continuous. We use our recent study of Bombay topologies to get the general solution and derive results in the case of various known hypertopologies. Sample results are:
(1) Let and be metric spaces and , be assigned the corresponding Hausdorff metric topologies. Then is uniformly continuous if, and only if, is (uniformly) continuous.
(2) Let and be topological spaces and , be assigned the corresponding Vietoris topologies. Then is continuous if, and only if, is continuous.
(3) Let and be Hausdorff topological spaces and , be assigned the corresponding Fell topologies. Then is continuous and strongly compact if, and only if, is continuous.