Proposition 4

Let $X$ be a set, let $(Y_i)_{i \in I}$ be a family of uniform spaces, and for each $i \in I$ let $f_i$ be a mapping of $X$ into $Y_i$. Let $X$ carry the coarsest uniformity $U$ for which the $f_i$ are uniformly continuous. Then in order that a filter base $\mathfrak{B}$ on $X$ should be a Cauchy filter base it is necessary and sufficient that $f_i(\mathfrak{B})$ should be a Cauchy filter base on $Y_i$, for each $i \in I$.

The condition is necessary by Proposition 3. Conversely, suppose that it is satisfied, and let $U(V_{i_1}, \ldots, V_{i_n})$ be an entourage of the uniformity $\mathfrak{U}$ [§ 2, no. 3, formula (1)]. By hypothesis, for each index $k$ there is a set $M_k \in \mathfrak{B}$ such that $f_{i_k}(M_k)$ is $V_{i_k}$-small ($1 \leq k \leq n$). Let $M$ be a set of $\mathfrak{B}$ contained in $M_k$ for $1 \leq k \leq n$; then for each pair of points $x, x'$ of $M$ we have $[f_{i_k}(x), f_{i_k}(x')] \in V_{i_k}$ for $1 \leq k \leq n$, so that

This completes the proof.