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The generalized compact-open topology  τ  C  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  G δ  -diagonal, then  τ  C  is Čech-complete, sieve complete and satisfies the  p -space property of Arhangel'skiǐ, respectively, if and only if  X  is Lindelöf and locally compact. Lindelöfness, paracompactness and normality of  τ  C  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 .

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