A Polish topology for the closed subsets of a Polish space

Beer, Gerald

doi:10.1090/S0002-9939-1991-1065940-6

A Polish topology for the closed subsets of a Polish space
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by Gerald Beer

Proc. Amer. Math. Soc. 113 (1991), 1123-1133

DOI: https://doi.org/10.1090/S0002-9939-1991-1065940-6

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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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