Topology of pointwise convergence

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

Definition

Suppose $X$ and $Y$ are topological spaces. Let $C(X,Y)$ denote the space of continuous maps from $X$ to $Y$. The topology of pointwise convergence on $C(X,Y)$ is defined in the following equivalent ways:

1. It is the natural topology such that convergence of a sequence of elements in the topology is equivalent to their pointwise convergence as functions.
2. It is the topology on $C(X,Y)$ arising as the subspace topology from the product topology on the space of all functions $Y^{X}$.

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