Let $X$ be a separable metric space and $Y$ be a second-countable $\sigma$-compact Hausdorff space. Then the compact-convergence (compact-open) topology on $C(Y,X)$ is metrizable with metric $$ d(f,g):= \sum_{n =0}^{\infty} \frac1{2^n} \sup_{y \in K_n} d_X(f(y),g(y)), $$ where $\{K_n\}_{n \in \mathbb{N}}$ is a countable compact cover of $Y$ and $d_X$ is the metric on $X$. Moreover, if $X$ is Banach then $C(Y,X)$ is a Fréchet space.

This is easy to show, but I'm looking for a reference to this result. Thanks in advance.