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Developable hyperspaces are metrizable. (English) Zbl 1066.54009
For a Hausdorff topological space $$X$$ let $$CL(X)$$ denote the hyperspace of all nonempty closed subsets of $$X$$ and $$K(X)$$ the hyperspace of all nonempty compact subsets of $$X$$. In the context of $$CL(X)$$ and $$K(X)$$ endowed with various topologies (as Vietoris topology, Fell topology, locally finite topology, bounded Vietoris topology) the following properties are studied: developability, metrizability, having a $$G_{\delta}$$-diagonal, submetrizability. Some characterization theorems are proved. For example, the following are equivalent: (i) $$CL(X)$$ with the Vietoris topology is Moore; (ii) $$CL(X)$$ with the Vietoris topology is developable; (iii) $$CL(X)$$ with the Vietoris topology is metrizable; (iv) $$X$$ is compact and metrizable.

##### MSC:
 54B20 Hyperspaces in general topology
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