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:
 * $V(K, U) = \left\{ f \in \mathcal C \left( X, Y \right) : f(K) \subset U \right\}$.

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

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