Let and be topological spaces. The set (often denoted also ) of continuous maps from to has a natural topology called the compact-open topology: a subbase of that topology consists of sets of the form , where is compact and is open?, which consists of all continuous maps such that .
If is a metric space then the compact-open topology is the topology of uniform convergence on compact subsets in the sense that in with the compact-open topology iff for every compact subset , uniformly on . If (in addition) the domain is compact then this is the topology of uniform convergence.
The compact-open topology is most sensible when the topology of is locally compact Hausdorff, for in this case with the compact-open topology is an exponential object in the category Top of all topological spaces. This implies the exponential law for spaces , i.e. the adjunction map is a bijection whenever is locally compact Hausdorff; and it becomes a homeomorphism if in addition is also Hausdorff. See also convenient category of topological spaces.
Ralph H. Fox, On Topologies for Function Spaces , Bull. AMS 51 (1945) pp.429-432. (pdf)