Let $M$ and $N$ be topological spaces. Are there necessary and sufficient conditions on the topological properties of the spaces such that $C(M,N)$ is metrizable?


For $M$ compact and $N$ a metric space, the space is obviously metrizable using the uniform convergence topology, $d(f,g)=\sup\_{x\in M}d(f(x),g(x))$.


And also, if $N$ is a metric space, but $M$ is not necessarily compact the space of continuous bounded functions $C\_0(M,N)=\{f\in C(M,N)\mid d(f(x),a)\leq K\_f, \forall x\in M\}$ for a point $a\in N$ and $K\_f>0$ is metrizable with the same distance.


But in general, which distances are usable in $C(M,N)$ in the context of a noncompact space $M$?



Metrization of spaces of functions
As to the compact-open topology of $C(X,Y)$, it is metrizable if and only if $Y$ is metrizable, and $X$ is [hemicompact](https://en.wikipedia.org/wiki/Hemicompact_space). 


