Uniformizability of the 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 is an element of CL(X) (= the family of all nonempty closed subsets of X) and f is a continuous function from B to Y. We show that if X and Y are Tychonoff spaces and X locally compact, then the generalized compact-open topology tau(C) on P is also a Tychonoff space. Moreover we find a useful description of a compatible uniformity for (P, tau(C)).