nLab
compact-open topology

Idea

A natural topology on mapping spaces of continuous functions.

Definition

Let $X$ and $Y$ be topological spaces. The set $\mathrm{Map}(X,Y)$ (often denoted also $C(X,Y)$) of continuous maps from $X$ to $Y$ has a natural topology called the compact-open topology: a subbase of that topology consists of sets of the form $U_{K,V}$, where $K\subset X$ is compact and $V\subset Y$ is open?, which consists of all continuous maps $f:X\to Y$ such that $f(K)\subset V$.

If $Y$ is a metric space then the compact-open topology is the topology of uniform convergence on compact subsets in the sense that $f_n\to f$ in $\mathrm{Map}(X,Y)$ with the compact-open topology iff for every compact subset $K\subset X$, $f_n\to f$ uniformly on $K$. If (in addition) the domain $X$ is compact then this is the topology of uniform convergence.

Properties

The compact-open topology is most sensible when the topology of $X$ is locally compact Hausdorff, for in this case $\mathrm{Map}(X,Y)$ with the compact-open topology is an exponential object $Y^X$ in the category Top of all topological spaces. This implies the exponential law for spaces , i.e. the adjunction map is a bijection $\mathrm{Top}(X,\mathrm{Map}(Y,Z))\cong \mathrm{Top}(X\times Y,Z)$ whenever $Y$ is locally compact Hausdorff; and it becomes a homeomorphism $\mathrm{Map}(X,\mathrm{Map}(Y,Z))\cong \mathrm{Map}(X\times Y,Z)$ if in addition $X$ is also Hausdorff. See also convenient category of topological spaces.

Related concepts

• topology of mapping spaces

• differential topology of mapping spaces

• C-k topology