# Topology of pointwise convergence

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. Let denote the space of continuous maps from to . The **topology of pointwise convergence** on is defined in the following equivalent ways:

- It is the natural topology such that convergence of a sequence of elements in the topology is equivalent to their pointwise convergence as functions.
- It is the topology on arising as the subspace topology from the product topology on the space of
*all*functions .

In particular, the *topology* of pointwise convergence is little influenced by the topology of , although the underlying *set* of the topology is influenced by .