The metrizability and completeness of the support-open topology on

This paper studies the metrizability and various kinds of completeness property of the space C s ( X ) of continuous real-valued functions on a Tychonoff space X, where the function space has the support-open topology. The metrizability of C s ( 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.