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