Corollary

Let $(X_i)_{i \in I}, (Y_i)_{i \in I}$ be two families of uniform spaces indexed by the same set $I$. For each $i \in I$, let $f_i$ be a mapping of $X_i$ into $Y_i$. If each of the $f_i$ is uniformly continuous, then so is the product mapping

Conversely, if the $X_i$ are non-empty and $f$ is uniformly continuous, then each $f_i$ is uniformly continuous.

f can be written as $x \to (f_i(\mathrm{pr}_i x))$, and the first part of the corollary therefore follows from Proposition 7. The second part is proved by considering a point $a = (a_i)$ of $\prod_{i \in I} X_i$ and repeating the argument of Chapter I, , no. 1, Corollary 1 to Proposition 1, with the phrase "continuous at $a$ (resp. $a_\kappa$)" replaced by "uniformly continuous"

The general criterion of transitivity of initial uniformities (no. 3, Proposition 5) shows that, as for the product of topological spaces (Chapter I, § 4, no. 1), the product of uniform spaces is associative and that the following is true: