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 $Y_i$. For each $i \in I$ let $g_i$ denote $f_i \times f_i$. Let $\mathfrak{S}$ be the set of subsets of $X \times X$ of the form $\overline{g_i}(V_i)$, where $i \in I$ and $V_i$ is an entourage of $Y_i$, and let $\mathfrak{B}$ be the set of all finite intersections

of sets of $\mathfrak{S}$. Then $\mathfrak{B}$ is a fundamental system of entourages of a uniformity $\mathfrak{U}$ on $X$ which is the initial uniform structure on $X$ with respect to the family $(f_i)$ (Set Theory, Chapter IV, § 2, no. 3), and in particular $\mathfrak{U}$ is the coarsest uniformity on $X$ for which all the mappings $f_i$ are uniformly continuous. But otherwise, let $h$ be a mapping of a uniform space $Z$ into $X$; then $h$ is uniformly continuous (when $X$ is endowed with the uniformity $\mathfrak{U}$) if and only if each of the mappings $f_i \circ h$ is uniformly continuous.

It is immediately seen that $\mathfrak{B}$ satisfies axioms $(B_1)$ and $(U'_1)$. If $W_i = \overline{g}_i^{-1}(V_i)$, then $\overline{W}_i = \overline{g}_i^{-1}(\overline{V}_i)$ and $\dot{W}_i = \dot{g}_i^{-1}(\dot{V}_i)$; hence $\mathfrak{B}$ also satisfies axioms $(U'_{II})$ and $(U'_{III})$ and is therefore a fundamental system of entourages of a uniformity $\mathfrak{U}$ on $X$. Furthermore, it follows immediately from the definition of $\mathfrak{U}$ and Definition 1 and no. 1 that $f_i$ is uniformly continuous for each index $i \in I$; hence (no. 1, Proposition 2) $f_i \circ h$ is uniformly continuous for each $i \in I$ if $h$ is. Conversely, suppose that $f_i \circ h$ is uniformly continuous for each $i \in I$, and consider a set $U(V_{i_1}, \ldots, V_{i_n})$; by hypothesis, for each $k$ such that $1 \leq k \leq n$, there is an entourage $W_k$ of $Z$ such that the relation $(z, z') \in W_k$ implies $[f_{i_k}(h(z)), f_{i_k}(h(z'))] \in V_k$; if

these $n$ relations are simultaneously satisfied whenever $z$ and $z'$ are $W$-close, so that we have then $(h(z), h(z')) \in U(V_{i_1}, \ldots, V_{i_n})$, and the proof is complete.