A study concerning splitting and jointly continuous topologies on $C(Y,Z)$

Let $Y$ and $Z$ be two fixed topological spaces and $C(Y,Z)$ the set of all continuous maps from $Y$ into $Z$. We construct and study topologies on $C(Y,Z)$ that we call ${\mathcal{F}}_n({\tau }_n)$-family-open topologies. Furthermore, we find necessary and sufficient conditions such that these topologies to be splitting and jointly continuous. Finally, we present questions concerning a further study on this area.