Proposition 3

Let $X$ be a set, let $(Y_\lambda)_{\lambda \in L}$ be a family of uniform spaces, and for each $\lambda \in L$ let $f_\lambda$ be a mapping of $X$ into $Y_\lambda$. Let $X$ carry the coarsest uniformity for which the $f_\lambda$ are uniformly continuous. Then a subset $A$ of $X$ is precompact if and only if $f_\lambda(A)$ is a precompact subset of $Y_\lambda$ for each $\lambda \in L$.

The condition is necessary by virtue of Proposition 2. Sufficiency follows from the characterization of the Hausdorff completion of $X$ given in § 3, no. 9, Proposition 18, and Tychonoff’s theorem (Chapter I, § 9, Theorem 3, Corollary).