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A natural functor for hyperspaces. (English) Zbl 1122.54006
Summary: Let $$(X,{\mathcal T})$$, $$(Y,{\mathcal T}')$$ 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 $$f:X\to Y$$ there is a natural function: $$F:CL(X)\to CL(Y)$$, where for each $$A\in CL(X)$$, $$F(A)=cl\,f (A)\in CL(Y)$$. 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:
 54B20 Hyperspaces in general topology 54C05 Continuous maps 54E05 Proximity structures and generalizations 54E15 Uniform structures and generalizations 54E35 Metric spaces, metrizability