nLab
compact-open topology

Let $X$ and $Y$ be topological spaces. The set $\mathrm{Map}(X,Y)$ (often denoted also $C(X,Y)$) of continuous maps from $X$ to $Y$ has a natural topology called the compact-open topology: a basis of that topology consists of sets of the form $U_{K,V}$, where $K\subset X$ is compact and $V\subset Y$ is open?, which consists of all continuous maps $f:X\to Y$ such that $f(K)\subset V$.

For the special case of the function spaces (that is when $Y$ is the real line or complex plane) where the domain is a metric space, the compact-open topology is also known as the topology of uniform convergence on compact subsets, or if the domain $X$ is also compact, as the topology of uniform convergence.

The compact-open topology is most sensible when the topology of $X$ is locally compact Hausdorff, for in this case $\mathrm{Map}(X,Y)$ with the compact-open topology is an exponential object $Y^X$ in the category $\mathrm{Top}$ of all topological spaces. This implies that we have an exponential law for spaces $\mathrm{Top}(X,\mathrm{Map}(Y,Z))\cong \mathrm{Top}(X\times Y,Z)$ whenever $Y$ is locally compact Hausdorff. See also convenient category of topological spaces.