The metrizability and completeness of the σ-compact-open topology on

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