Im trying to obtain the less restrictive conditions on the topological properties on $M$, $N$, (Combinations of second- countable, lindelöf, hausdorff, paracompact or some othe properties) such that $C(M,N)$ is metrizable with some dynamically relevant topology in the sense that the stability notion of dynamical systems (en.wikipedia.org/wiki/Topological_conjugacy) is not vacuous.