Polishness of Weak Topologies Generated by Gap and Excess Functionals

Let (X, d) be a separable complete metric space and CL(X) the family of all nonempty closed subsets of X. We show that the finite Hausdorff topology on CL(X) is Polish. The finite Hausdorff topology is measurably compatible on CL(X) [1], i.e. its Borel field coincides with the Effros sigma algebra. So Polishness of this topology can be useful for measurable multifunctions with values in CL(X) equipped with the finite Hausdorff topology. Polishness of other weak topologies generated by a family of gap and excess funtionals is also proved.