\begin{definition}[Definition:Compact-Open Topology]
Let $X$ and $Y$ be topological spaces.
Let $\map \CC {X, Y}$ 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:
:$\map V {K, U} = \set {f \in \map \CC {X, Y}: f \sqbrk K \subset U}$
Let:
:$\BB = \set {\map V {K, U}: K \subset X \text{ compact}, U \subset Y \text{ open} }$
The '''compact-open topology''' on $\map \CC {X, Y}$ is the topology generated by $\BB$.
\end{definition}