The metrizability and completeness of the σ-compact-open topology on C⁎(X)

This paper studies the metrizability and various kinds of completeness properties of the space of continuous real-valued bounded functions on a Tychonoff space X, where the function space has the σ-compact-open topology. The metrizability of 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 are studied.