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