Definition 5.3.3 (Pseudometric Uniformity). Let $X$ be a set and $\seqi{d}$ be pseudometric on $X$. For each $i \in I$, $r > 0$, and $x \in X$, let
and
then there exists a uniformity $\fU$ on $X$ such that:
The family
\[\fB = \bracs{\bigcap_{j \in J}E(d_j, r) \bigg | J \subset I \text{ finite}, r > 0}\]forms a fundamental system of entourages consisting of symmetric open sets.
For any $x \in X$,
\[\cb(x) = \bracs{\bigcap_{j \in J}B_j(x, r) \bigg | J \subset I \text{ finite}, r > 0}\]is a fundamental system of neighbourhoods at $x$.
For each $U \subset X$, $U$ is open if and only if for every $x \in U$, there exists $J \subset I$ finite and $r > 0$ such that $\bigcap_{j \in J}B_{j}(x, r) \subset U$.
For each $i \in I$, $d_{i} \in UC(X \times X; [0, \infty))$.
For any other uniformity $\mathfrak{V}$ satisfying (4), $\mathfrak{U}\subset \mathfrak{V}$.
The uniformity $\fU$ is the pseudometric uniformity induced by $\seqi{d}$, and the topology induced by $\fU$ is the pseudometric topology on $X$ induced by $\seqi{d}$.
Proof. (1, fundamental system): To see that $\fB$ is a fundamental system of entourages for a uniformity on $X$, it is sufficient to verify the conditions of Proposition 5.1.8.
For any $J, J' \subset I$ finite and $r, r' > 0$,
\[\bigcap_{j \in J \cup J'}E(d_{j}, \min(r,r')E(d_{j}, r \wedge r') \subset \paren{\bigcap_{j \in J}E(d_j, r)}\cap \paren{\bigcap_{j \in J'}E(d_j, r')}\]For each $i \in I$, $d(x, x) = 0$ for all $x \in X$. Thus for any $i \in I$ and $r > 0$, $E(d_{i}, r)$ contains the diagonal.
For each $J \subset I$ finite and $r > 0$,
\[\paren{\bigcap_{j \in J}E(d_j, r/2)}\circ \paren{\bigcap_{j \in J}E(d_j, r)}\subset \bigcap_{j \in J}E(d_{j}, r/2) \circ E(d_{j}, r/2) \subset \bigcap_{j \in J}E(d_{j}, r)\]by the triangle inequality.
(2): Since $\fB$ is a fundamental system of entourages for $\fU$,
is a fundamental system of neighbourhoods at $x$.
(3): By definition of the uniform topology, for any $U \subset X$, $U$ is open if and only if for any $x \in U$, there exists $V \in \fU$ such that $x \in V(x) \subset U$. As $\fB$ is a fundamental system of entourages for $\fU$, this is equivalent to the existence of $J \subset I$ finite and $r > 0$ such that
(1, symmetric open): Let $i \in I$ and $r > 0$. Since $d_{i}$ is symmetric, so is $E(d_{i}, r)$. For any $(x, y) \in E(d_{i}, r)$, let $s = r - d_{i}(x, y)$, then for any $(x', y') \in X \times X$ with $d_{i}(x, x') < s/2$ and $d_{i}(y, y') < s/2$, $d(x', y') < s + d_{i}(x, y) = r$. Thus $B_{i}(x, s/2) \times B_{i}(y, s/2) \subset E(d_{i}, r)$.
By (3), $B_{i}(x, s/2) \in \cn(x)$ and $B_{i}(y, s/2) \in \cn(y)$, so $B_{i}(x, s/2) \times B_{i}(y, s/2) \in \cn((x, y))$. As such, $E(d_{i}, r)$ is open by Lemma 4.4.3.
(4): By Lemma 5.3.2.
(5): By Lemma 5.3.2, $E(d_{i}, r) \in \mathfrak{V}$ for all $i \in I$ and $r > 0$, so $\mathfrak{V}\supset \mathfrak{B}$ by (F2), and $\mathfrak{V}\supset \mathfrak{U}$.$\square$