---
tag: V1PAXEF0EE
kind: definition
name: Compact-open topology
msc: [54C35]
defines: RG15F0TWPW
uses: [0CQHFPGJ45, 0VGFHTM75Y, 0R6EWFKRCF, 02BH3MCBQB, 09S8DV34Y0]
sources:
  - url: https://en.wikipedia.org/wiki/Compact-open_topology
---

## Definition

Let $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]], and let $C(X, Y)$ be the set of [[continuous maps|0VGFHTM75Y]] from $X$ to $Y$.
For a [[compact|0R6EWFKRCF]] subset $K \subseteq X$ and an [[open|02BH3MCBQB]] subset $U \subseteq Y$, put
$$V(K, U) = \set{f \in C(X, Y) : f(K) \subseteq U}.$$
The *compact-open topology* on $C(X, Y)$ is the topology generated by the [[subbasis|09S8DV34Y0]] of all sets $V(K, U)$.

## Remarks

If $(Y, d)$ is a metric space, the compact-open topology is the topology of uniform convergence on compact subsets of $X$.
In particular, a sequence $(f_n)$ converges to $f$ if and only if $\sup_{x \in K} d(f_n(x), f(x)) \to 0$ for every compact $K \subseteq X$.
If $X$ is locally compact Hausdorff, the evaluation map $C(X, Y) \times X \to Y$, $(f, x) \mapsto f(x)$, is continuous.

Example: for $X = [0, 1]$ and $Y = \R$, the space $X$ is compact, so convergence in $C([0, 1], \R)$ is uniform convergence on $[0, 1]$.

Non-example: the topology of pointwise convergence on $C([0, 1], \R)$ is not the compact-open topology.
Let $f_n$ be the piecewise linear map with $f_n(0) = 0$, $f_n(1/n) = 1$ and $f_n(x) = 0$ for $x \ge 2/n$.
Then $f_n(x) \to 0$ for each $x$, but $f_n \notin V([0, 1], (-1, 1))$ for every $n$, so $f_n$ does not converge to $0$ in the compact-open topology.