Compact-open topology

Let $X$ and $Y$ be topological spaces, and let $C(X,Y)$ be the space of continuous functions from $X$ to $Y.$ Given $K\subset X$ and $U\subset Y$, define a subset of $C(X,Y)$: $$O_{K,U} = \{ f\in C(X,Y):\: f(K)\subset U\} .$$ Define the compact-open topology on $C(X,Y)$ to be the topology generated by the sub-basis $$\gamma =\{ O_{K,U}:\: K\subset X\,\text{compact,}\quad U\subset Y\, \text{open} \}.$$

If $Y$ is a metric space, then this topology coincides with the topology of uniform convergence on compact sets. If $X$ is a compact space, this is the topology of uniform convergence.