compact-open topology

Let $X$ and $Y$ be topological spaces, and let $C(X,Y)$ be the set of continuous maps from $X$ to $Y.$ Given a compact subspace $K$ of $X$ and an open set $U$ in $Y,$ let

$${𝒰}_{K,U}:=\{f\in C(X,Y):f(x)\in U\text{whenever}x\in K\}.$$

Define the compact-open topology on $C(X,Y)$ to be the topology generated by the subbasis

$$\{{𝒰}_{K,U}:K\subset X\text{compact,}\mathit{\hspace{1em}}U\subset Y\text{open}\}.$$

If $Y$ is a uniform space (for example, if $Y$ is a metric space), then this is the topology of uniform convergence on compact sets. That is, a sequence $\left(f_n\right)$ converges to $f$ in the compact-open topology if and only if for every compact subspace $K$ of $X,$ $\left(f_n\right)$ converges to $f$ uniformly on $K$. If in addition $X$ is a compact space, then this is the topology of uniform convergence.