Properties of the $C$-compact-open topology on a function space

We study the $C$-compact-open topology on the set $C(X)$ of all continuous real-valued functions defined on a Tikhonov space $X$. The relations between the $C$-compact-open topology and the compact-open and bounded-open topologies on the set $X$ are studied. We also investigate the cardinal-valued characteristics of the space $C(X)$ equipped with the $C$-compact-open topology, for example, the Suslin number, Lindelöf number, weight, and density.