# Compact-open topology

From Topospaces

*This article defines a function space topology i.e. a topology on the collection of continuous maps between two topological spaces*

## Definition

Suppose and are topological spaces. The **compact-open topology** is a topology we can define on the space of continuous functions from to as follows.

For a compact subset and an open subset , we define as the set of all continuous maps such that . The compact-open topology is the topology with subbasis as the set of all s.

## Relation with other function space topologies

Topology | Meaning | Relationship with compact-open topology |
---|---|---|

topology of pointwise convergence | topology chosen such that a sequence of functions converges iff it converges pointwise; equivalently, the subspace topology inherited from the product topology on the space of all functions. | ? |

topology of uniform convergence | ||

topology of compact convergence |