Proposition 10

Every product of complete uniform spaces is complete. Conversely, if a product of non-empty uniform spaces is complete, then each of the factors is a complete uniform space.

The first assertion is a consequence of the characterization of Cauchy filters and convergent filters on a product space (no. 1, Corollary 2 to Proposition 4 and Chapter I, § 7, no. 6, Corollary 1 to Proposition 10). Conversely, suppose

is complete (the $X_i$ being non-empty) and let $\mathfrak{F}_x$ be a Cauchy filter on $X_x$. For each $i \neq x$ let $\mathfrak{F}_i$ be a Cauchy filter on $X_i$, and consider the product filter (Chapter I, § 6, no. 7)

$\mathfrak{F}$ is a Cauchy filter (no. 1, Corollary 2 to Proposition 4), hence is convergent, and therefore so is each $\operatorname{pr}_x \mathfrak{F} = \mathfrak{F}_x$ (Chapter I, § 7, no. 6, Corollary 1 to Proposition 10).