Topology of uniform convergence

Definition

Suppose $X$ is a topological space and $Y$ is a uniform space (for instance, $Y$ may be a metric space or a $T_{0}$-topological group). Let $C(X,Y)$ denote the space of continuous functions from $X$ to $Y$. Then, we can define a topology of uniform convergence on $C(X,Y)$ as follows