Proposition 8

Let $X$ be a set, let $(Y_i)_{i \in I}$ be a family of uniform spaces, and for each $i \in I$ let $f_i$ be a mapping of $X$ into $Y_i$. Let $f$ be the mapping $x \to (f_i(x))$ of $X$ into $Y = \prod_{i \in I} Y_i$, and let $U$ be the coarsest uniformity on $X$ for which the $f_i$ are uniformly continuous. Then $U$ is the inverse image under $f$ of the uniformity induced on $f(X)$ by the product uniformity on $Y$.