= Compact-open topology
{wiki}
For <topological spaces> $X,Y$, the compact-open topology on $C(X,Y)$ has subbasic sets
$$
[K,U]=\{f\in C(X,Y):f(K)\subseteq U\},
$$
where $K\subseteq X$ is <compact> and $U\subseteq Y$ is <open>. Finite intersections of such conditions describe neighborhoods of a <continuous map>. This definition does not require a metric on either space.
For a metric target and <continuous functions>, it is the topology of uniform convergence on compact subsets. In the scalar case, the compact <seminorms> $p_K(f)=\sup_{x\in K}|f(x)|$ describe it. If $X$ is a <locally compact space> and a <Hausdorff space>, this compact control also agrees with <locally uniform convergence>, because each point has a compact neighborhood.
On $\mathbb R$, control on every interval $[-n,n]$ is sufficient. It permits functions such as translated compactly supported bumps to converge to zero on every compact set while retaining the same global <supremum norm>. The compact-open topology records local approximation rather than a bound over an unbounded domain all at once.
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