Corollary

Let $X$ be a topological space and let $Y$ be a metrizable uniform space. Suppose that there is a sequence $(K_n)$ of compact subsets of $X$ such that every compact subset of $X$ is contained in some $K_n$. Then the uniformity of compact convergence on $\mathscr{F}(X; Y)$ is metrizable.

Since the $K_n$ cover $X$, $\mathscr{F}_c(X; Y)$ is isomorphic to a uniform subspace of $\prod_n \mathscr{F}_u(K_n; Y)$ ($§ 1$, no. 2, Remark 3), and the corollary therefore follows from Proposition 1 (Chapter IX, § 2, no. 4, Theorem 1, Corollary 2).

Note that this corollary applies in particular if $X$ is locally compact and σ-compact (Chapter I, § 9, no. 9, Proposition 15, Corollary 1).

Now let $Y$ be a metric space and let $d$ be its metric. If $X$ is any set and $\mathscr{S}$ any set of subsets of $X$, we shall denote by $\mathscr{B}_{\mathscr{S}}(X; Y)$ the set of all mappings $u : X \to Y$ such that $u(A)$ is bounded for each $A \in \mathscr{S}$. Unless the contrary is expressly stated we shall regard $\mathscr{B}_{\mathscr{S}}(X; Y)$ as endowed with the uniformity of $\mathscr{S}$-convergence, which is defined by the following family of pseudometrics on $\mathscr{B}_{\mathscr{S}}(X; Y)$:

which are finite by hypothesis. When $\mathscr{S} = \{X\}$, we write $\mathscr{B}(X; Y)$ in place of $\mathscr{B}_{\mathscr{S}}(X; Y)$. A mapping $u : X \to Y$ is said to be bounded if it belongs to $\mathscr{B}(X; Y)$, i.e. if $u(X)$ is a bounded subset of $Y$.