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A natural functor for hyperspaces. (English) Zbl 1122.54006

Summary: Let (X,𝒯), (Y,𝒯 ' ) be Hausdorff topological spaces and CL(X), CL(Y) respectively the families of all non empty closed subsets of X and Y with some hypertopologies assigned. For each function f:XY there is a natural function: F:CL(X)CL(Y), where for each ACL(X), F(A)=clf(A)CL(Y). In this paper, we study the relationship between f and F. We are primarily interested in finding necessary and sufficient conditions on f to ensure the continuity of F. To avoid trivial situations, we will assume that Y contains at least two points and an arc and f is a surjection. Since the base spaces X,Y are embedded in their hyperspaces, we always have f 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 X and Y be metric spaces and CL(X), CL(Y) be assigned the corresponding Hausdorff metric topologies. Then f is uniformly continuous if, and only if, F is (uniformly) continuous.

(2) Let X and Y be topological spaces and CL(X), CL(Y) be assigned the corresponding Vietoris topologies. Then f is continuous if, and only if, F is continuous.

(3) Let X and Y be Hausdorff topological spaces and CL(X), CL(Y) be assigned the corresponding Fell topologies. Then f is continuous and strongly compact if, and only if, F is continuous.

MSC:
54B20Hyperspaces (general topology)
54C05Continuous maps
54E05Proximity structures and generalizations
54E15Uniform structures and generalizations
54E35Metric spaces, metrizability