Definition 4

If $(X_i)_{i \in I}$ is a family of uniform spaces, the product uniform space of this family is the product set

endowed with the coarsest uniformity for which the projections $pr_i : X \to X_i$ are uniformly continuous. This uniformity is called the product of the uniformities of the $X_i$, and the uniform spaces $X_i$ are called the factors of $X$.

The topology induced by the product uniformity on $X$ is same as the product of the topologies of the $X_i$ (no. 3, Corollary to Proposition 4).