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A natural functor for hyperspaces. (English) Zbl 1122.54006

Summary: Let $\left(X,𝒯\right)$, $\left(Y,{𝒯}^{\text{'}}\right)$ be Hausdorff topological spaces and $CL\left(X\right)$, $CL\left(Y\right)$ 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: $F:CL\left(X\right)\to CL\left(Y\right)$, where for each $A\in CL\left(X\right)$, $F\left(A\right)=cl\phantom{\rule{0.166667em}{0ex}}f\left(A\right)\in CL\left(Y\right)$. 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 $CL\left(X\right)$, $CL\left(Y\right)$ 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 $CL\left(X\right)$, $CL\left(Y\right)$ 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 $CL\left(X\right)$, $CL\left(Y\right)$ be assigned the corresponding Fell topologies. Then $f$ is continuous and strongly compact if, and only if, $F$ is continuous.

##### MSC:
 54B20 Hyperspaces (general topology) 54C05 Continuous maps 54E05 Proximity structures and generalizations 54E15 Uniform structures and generalizations 54E35 Metric spaces, metrizability