The metrizability and completeness of the support-open topology on C(X)

This paper studies the metrizability and various kinds of completeness property of the space Cs(X) of continuous real-valued functions on a Tychonoff space X, where the function space has the support-open topology. The metrizability of Cs(X) is studied in the context of generalized metric spaces and generalizations of the first countability. On the other hand the completeness properties, studied here, range from complete metrizability to pseudocompleteness.