Remark 3

Let $X$ be a locally compact space and $Y$ a Hausdorff topological space. If $\mathscr{T}$ is a topology on a subset $H$ of $C(X; Y)$ such that the mapping $(u, x) \to u(x)$ is continuous on $H \times X$ and if also $H$ is compact with respect to $\mathscr{T}$, then $\mathscr{T}$ is the compact-open topology. For it is finer than the latter by Corollary 1, and since the compact-open topology is Hausdorff, the two topologies are identical. Note that if in addition $Y$ is completely regular, then $H$ is equicontinuous with respect to every uniformity compatible with the topology of $Y$ (§ 2, no. 5, Theorem 2, Corollary 3), and for every compact subset $K$ of $X$ the set

is compact, since it is the image of $H \times K$ under the continuous mapping $(u, x) \to u(x)$.