Corollary

For each $i \in I$, let $A_i$ be a subspace of $Y_i$. Then the uniformity induced on $A = \prod_{i \in I} A_i$ by the product uniformity on $\prod_{i \in I} Y_i$ is the same as the product of the uniformities of the subspaces $A_i$.

In addition, we see immediately that if $X_1, X_2$ are two uniform spaces and $a_1$ is any point of $X_1$, the mapping $x_2 \to (a_1, x_2)$ is an isomorphism of $X_2$ onto the subspace $\{a_1\} \times X_2$ of $X_1 \times X_2$; hence: