## You are here

Homecompact-open topology

## Primary tabs

# 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

## Mathematics Subject Classification

54-00*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith

Sep 17

new question: Harshad Number by pspss

Sep 14

new problem: Geometry by parag

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith

Sep 17

new question: Harshad Number by pspss

Sep 14

new problem: Geometry by parag

## 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