 Compact-open topology

For topological spaces $X,Y$, the compact-open topology on $C(X,Y)$ has subbasic setswhere $K\subseteq X$ is compact and $U\subseteq Y$ is open. Finite intersections of such conditions describe neighborhoods of a continuous map. This definition does not require a metric on either space.

For a metric target and continuous functions, it is the topology of uniform convergence on compact subsets. In the scalar case, the compact seminorms $p_K(f)={\sup }_{x\in K}|f(x)|$ describe it. If $X$ is a locally compact space and a Hausdorff space, this compact control also agrees with locally uniform convergence, because each point has a compact neighborhood.

On $\mathbb{R}$, control on every interval $[-n,n]$ is sufficient. It permits functions such as translated compactly supported bumps to converge to zero on every compact set while retaining the same global supremum norm. The compact-open topology records local approximation rather than a bound over an unbounded domain all at once.

The compact-open topology is a topology defined on the space of continuous functions between topological spaces, particularly when considering the set of continuous functions from one topological space to another. This topology is especially useful in areas like functional analysis and algebraic topology.