Definition 5.2.3 (Initial Uniformity). Let $X$ be a set, $\bracsn{(Y_i, \fU_i)}_{i \in I}$ be a family of uniform spaces, and $\seqi{f}$ be a family of maps such that $f_{i}: X \to Y_{i}$ for each $i \in I$, then there exists a uniformity $\fU$ on $X$ such that:
For each $i \in I$, $f_{i} \in UC(X; Y_{i})$.
If $\mathfrak{V}$ is a uniformity on $X$ satisfying $(1)$, then $\mathfrak{V}\supset \fU$.
Moreover,
The family
\[\fB = \bracs{\bigcap_{j \in J}(f_j \times f_j)^{-1}(U_j) \bigg | J \subset I \text{ finite}, U_j \in \fU_j}\]is a fundamental system of entourages for $\fU$.
For any uniform space $Y$ and map $f: Y \to X$, $f \in UC(Y; X)$ if and only if $f_{i} \circ f \in UC(Y; Y_{i})$ for all $i \in I$.
known as the initial uniformity on $X$ generated by $\seqi{f}$.
Proof. (3): Since the diagonal is mapped to the diagonal and $\fB$ is closed under intersections, it is sufficient to verify (UB3) for $\fB$. Let $J \subset I$ be finite and $\bigcap_{j \in J}(f_{j} \times f_{j})^{-1}(U_{j}) \in \fB$, then there exists $\bracs{V_j}_{j \in J}$ such that $V_{j} \circ V_{j} \subset U_{j}$ for each $j \in J$. In which case, for any $(x, y), (y, z) \in f_{j}^{-1}(V_{j})$, $(f(x), f(y)), (f(y), f(z)) \in V_{j}$ and $(f(x), f(z)) \in U_{j}$. Thus $(f_{j} \times f_{j})^{-1}(V_{j}) \circ (f_{j} \times f_{j})^{-1}(V_{j}) \subset (f_{j} \times f_{j})^{-1}(U_{j})$, and
By Proposition 5.1.8, there exists a uniformity $\fU$ such that $\fB$ is a fundamental system of entourages for $\fU$.
(1): $\fU \supset (f_{i} \times f_{i})^{-1}(\fU_{i})$ for all $i \in I$.
(U): For any $i \in I$, $\mathfrak{V}\supset (f_{i} \times f_{i})^{-1}(\fU_{i})$. By (F2), $\mathfrak{V}\supset \fB$, so $\mathfrak{V}\supset \fU$.
(4): Let $J \subset I$ finite and $\seqj{U_j}$ such that $U_{j} \in \fU_{j}$ for each $j \in J$, then
is an entourage of $Y$.$\square$