On the completeness properties of the $C$-compact-open topology on $C(X)$

The completeness properties of the space $C_{rc}(X)$ of all continuous real-valued functions on a Tikhonov space $X$ are studied, where the function space has the $C$-compact-open topology. The properties of complete metrizability, Čech completeness, pseudocompleteness, and almost Čech-comp leteness are investigated.