Definition 1

Let $X, Y$ be two topological spaces, not necessarily uniformizable. For each compact subset $K$ of $X$ and each open subset $U$ of $Y$, let $T(K, U)$ be the set of all $u \in C(X; Y)$ such that $u(K) \subset U$. The topology on $C(X; Y)$ generated by the sets $T(K, U)$ is called the topology of compact convergence or the compact-open topology; and we denote by $C_c(X; Y)$ the topological space obtained by endowing $C(X; Y)$ with this topology.

If $Y$ is a uniform space it follows from Theorem 2 that this definition agrees with that given in § 1, no. 3.

If $H$ is a subset of $C(X; Y)$ we shall say that the topology induced on $H$ by that of $C_c(X; Y)$ is the compact-open topology on $H$.