Definition 6.2.1 (Uniform Topology). Let $T$ be a set and $(X, \fU)$ be a uniform space. For each $U \in \fU$, let
\[E(U) = \bracsn{(f, g) \in X^T| (f(x), g(x)) \in U \forall x \in X}\]
then the uniform topology on $X^{T}$ is the topology induced by the uniformity generated by $\bracs{E(U)| U \in \fU}$.