Exercise 5, § 3

Let $X$ be a completely regular space. a) If every point of the space $\mathscr{C}_c(X; \mathbf{R})$ has a countable fundamental system of neighbourhoods, then there exists an increasing sequence $(K_n)$ of compact subsets of $X$ such that every compact subset of $X$ is contained in some $K_n$; and $\mathscr{C}_c(X; Y)$ is then metrizable for any metrizable space $Y$. b) $\mathscr{C}_c(X; \mathbf{R})$ is metrizable and of countable type if and only if all the compact subspaces $K_n$ are metrizable (use Exercise 4). For every metrizable space $Y$ of countable type, $\mathscr{C}_c(X; Y)$ is then metrizable and of countable type.