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# 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

${\mathcal{U}}_{{K,U}}:=\left\{f\in C(X,Y):\>f(x)\in U\,\text{whenever}\,x\in K% \right\}.$ |

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

$\left\{{\mathcal{U}}_{{K,U}}:\>K\subset X\,\text{compact,}\quad U\subset Y\,% \text{open}\right\}.$ |

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.

Related:

UniformConvergence

Synonym:

topology of compact convergence

Type of Math Object:

Definition

Major Section:

Reference

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## Comments

## References

Hi,

Could you please post reference where it is shown that the topology coincide with the topology of uniform convergence in the case X is compact and Y is a uniform space. Thanks