Completeness properties of the generalized compact-open topology on partial functions with closed domains

The primary goal of the paper is to investigate the Baire property and weak α-favorability for the generalized compact-open topology τC on the space P of continuous partial functions f :A→Y with a closed domain A⊂X. Various sufficient and necessary conditions are given. It is shown, e.g., that (P,τC) is weakly α-favorable (and hence a Baire space), if X is a locally compact paracompact space and Y is a regular space having a completely metrizable dense subspace. As corollaries we get sufficient conditions for Baireness and weak α-favorability of the graph topology of Brandi and Ceppitelli introduced for applications in differential equations, as well as of the Fell hyperspace topology. The relationship between τC, the compact-open and Fell topologies, respectively is studied; moreover, a topological game is introduced and studied in order to facilitate the exposition of the above results.