Proposition 5.2.2. Let $(X, \fU)$ and $(Y, \mathfrak{V})$ be uniform spaces and $f \in UC(X; Y)$, then $f \in C(X; Y)$.
Proof. Let $x \in X$ and $V(f(x)) \in \cn(f(x))$, then since $f^{-1}(V(f(x))) =[f^{-1}(V)](x)$, by (2) of Definition 5.2.1, $f^{-1}(V(f(x))) \in \cn(x)$. As this holds for all $x \in X$, $f$ is continuous.$\square$