A Polish topology for the closed subsets of a Polish space

Abstract

Let ⟨ X , d ⟩ \left \langle {X,d} \right \rangle be a complete and separable metric space. The Wijsman topology on the nonempty closed subset CL ⁡ ( X ) \operatorname {CL}\left ( X \right ) of X X is the weakest topology on CL ⁡ ( X ) \operatorname {CL}\left ( X \right ) such that for each x x in X X , the distance functional A → d ( x , A ) A \to d\left ( {x,A} \right ) is continuous on CL ⁡ ( X ) \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 CL ⁡ ( X ) \operatorname {CL}\left ( X \right ) , equipped with Wijsman topology.