  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 ${\cal F}_n(\tau_n)$-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.
