Definition:Compact-Open Topology

Definition
Let $X$ and $Y$ be topological spaces.

Let $\mathcal C \left({X, Y}\right)$ be the set of continuous maps from $X$ to $Y$.

For all compact subsets $K \subset X$ and all open subsets $U \subset Y$, let:
 * $V \left({K, U}\right) = \left\{ {f \in \mathcal C \left({X, Y}\right): f \left({K}\right) \subset U}\right\}$

Let:
 * $\mathcal B = \left\{ {V \left({K, U}\right): K \subset X \text{ compact}, U \subset Y \text{ open} }\right\}$

The compact-open topology on $\mathcal C \left({X, Y}\right)$ is the topology generated by $\mathcal B$.

Also see

 * Compact-Open Topology is Topology which proves that this is in fact a topology.