Proposition 5.2.4. Let $X$ be a set, $\bracsn{(Y_i, \fU_i)}_{i \in I}$ be a family of uniform spaces, and $\seqi{f}$ where $f_{i}: X \to Y_{i}$ for each $i \in I$, then the initial topology $\topo$ on $X$ coincides with the topology $\topo_{U}$ induced by the initial uniformity.
Proof. By Proposition 5.2.2 and (1) of Definition 5.2.3, $f_{i}: X \to Y_{i}$ is continuous with respect to $\topo_{U}$ for all $i \in I$. By (U) of Definition 4.1.7, $\topo_{U} \supset \topo$.
On the other hand, let $x \in X$ and $U \in \cn_{\topo_U}(x)$, then there exists an entourage $V$ such that $U = V(x)$. By (3) of Definition 5.2.3, there exists $J \subset I$ finite and $\seqj{V}$ such that $\bigcap_{j \in J}(f_{j} \times f_{j})^{-1}(V_{j}) \subset V$ and each $V_{j} \in \fU_{j}$. In which case, $f_{j}^{-1}(V_{j}(f_{j}(x))) \in \cn(x)$ for all $j \in J$. Using (F2),
where for any $y \in W$, $(f_{j}(x), f_{j}(y)) \in V_{j}$ for all $j \in J$. Thus $f(y) \in V(x) = U$, and $\topo_{U} \subset \topo$.$\square$