Topology of pointwise convergence

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 $$X$$.