A natural functor for hyperspaces.

*(English)*Zbl 1122.54006Summary: 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.

(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 |