Proposition 6.1.3. Let $T$ be a set, $\mathfrak{S}\subset 2^{T}$ be a non-empty family of sets, and $(X, \fU)$ be a uniform space whose uniformity is induced by the pseudometrics $\seqi{d}$. For each $i \in I$ and $S \in \mathfrak{S}$, let
then
$\bracs{d_{i, S}| i \in I, S \in \mathfrak{S}}$ is a family of pseudometrics induces the $\mathfrak{S}$-uniformity on $X^{T}$.
If $\mathfrak{S}$ is upward-directed with respect to inclusion, then
\[\bracs{\bigcap_{j \in J}E(d_{j, S}, r)|J \subset I \text{ finite}, r > 0, S \in \mathfrak{S}}\]is a fundamental system of entourages for the $\mathfrak{S}$-uniformity on $X^{T}$.
Proof. (1): Let $S \in \mathfrak{S}$ and $U \in \fU$, then there exists $r > 0$ and $J \subset I$ finite such that $\bigcap_{j \in J}E(d_{j}, r) \subset U$, so
and the uniformity induced by $\bracs{d_{i, S}| i \in I, S \in \mathfrak{S}}$ contains the $\mathfrak{S}$-uniformity.
On the other hand, for any $i \in I$ and $r > 0$, $E(d_{j}, r/2) \in \fU$ by Definition 5.3.3. Therefore $E(S, E(d_{j}, r/2)) \subset E(d_{j, S}, r)$, so the $\mathfrak{S}$-uniformity contains the induced uniformity.
(2): If $\mathfrak{S}$ is upward-directed with respect to inclusion, then by Definition 6.1.2,
Following the same steps in (1),
is a fundamental system of entourages for the $\mathfrak{S}$-uniformity.$\square$