Cech-completeness and related properties of the generalized compact-open topology

The generalized compact-open topology tau(C) on partial continuous functions with closed domains in X and values in Y is studied. If Y is a non-countably compact Cech-complete space with a G(delta)-diagonal, then tau(C) is Cech-complete, sieve complete and satisfies the p-space property of Arhangel'skil, respectively, if and only if X is Lindelof and locally compact. Lindelofness, paracompactness and normality of tau(C) is also investigated. New results are obtained on Cech-completeness, sieve completeness and the p-space property for the compact-open topology on the space of continuous functions with a general range Y