Complete metrizability of generalized compact-open topology

Let X and Y be Hausdorff topological spaces. Let P be the family of all partial maps from X to Y; a partial map is a pair (B, f), where B ϵ CL(X) (= the family of all nonempty closed subsets of X) and f is a continuous function from B to Y. Denote by τc the generalized compact-open topology on P. We show that if X is a hemicompact metrizable space and Y is a Fréchet space, then (P, τC) is completely metrizable and homeomorphic to a closed subspace of (CL(X), τF) × (C(X,Y), τCO), where τF is the Fell topology on CL(X) and τCO is the compact-open topology on C(X,Y).