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A Polish topology for the closed subsets of a Polish space

Author: Gerald Beer
Journal: Proc. Amer. Math. Soc. 113 (1991), 1123-1133
MSC: Primary 54B20; Secondary 54C35, 54C60, 54E50
MathSciNet review: 1065940
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Abstract: Let $ \left\langle {X,d} \right\rangle $ be a complete and separable metric space. The Wijsman topology on the nonempty closed subset $ \operatorname{CL}\left( X \right)$ of $ X$ is the weakest topology on $ \operatorname{CL}\left( X \right)$ such that for each $ x$ in $ X$, the distance functional $ A \to d\left( {x,A} \right)$ is continuous on $ \operatorname{CL}\left( X \right)$. We show that this topology is Polish, and that the traditional extension of the topology to include the empty set among the closed sets is also Polish. We also compare the Borel class of a closed valued multifunction with its Borel class when viewed as a single-valued function into $ \operatorname{CL}\left( X \right)$, equipped with Wijsman topology.

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Keywords: Polish space, hyperspace, Wijsman topology, distance functional, topology of pointwise convergence, multifunction
Article copyright: © Copyright 1991 American Mathematical Society

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