Definition 6.1.2 (Set Uniformity). Let $T$ be a set, $\mathfrak{S}\subset 2^{T}$ be a non-empty family of sets, and $(X, \fU)$ be a uniform space. For each $S \in \mathfrak{S}$ and $U \in \fU$, let
and
then
$\mathfrak{E}(\mathfrak{S}, \fU)$ generates a uniformity $\fV$ on $X^{T}$.
The topology induced by $\fV$ is finer than the $\mathfrak{S}$-topology on $T^{X}$.
If $\mathfrak{S}$ is upward-directed with respect to inclusion, then $\mathfrak{E}(\mathfrak{S}, \fU)$ is forms a fundamental system of entourages for $\fV$.
The uniformity $\fV$ is the $\mathfrak{S}$-uniformity, and the topology induced by $\fV$ is the topology of uniform convergence on the sets $\mathfrak{S}$/$\mathfrak{S}$-uniform topology on $X^{T}$.
Proof. (1): Since $\Delta \subset E(S, U)$ for all $S \in \mathfrak{S}$ and $U \in \fU$, $\mathfrak{E}(\mathfrak{S}, \fU)$ generates a uniformity on $X^{T}$.
(2): Let $U \subset X$ be open, then for each $x \in U$, there exists $V_{x} \in \fU$ such that $x \in V_{x}(x) \subset U$. In which case, $U = \bigcup_{x \in U}V_{x}(x)$ and $M(S, U) = \bigcup_{x \in U}M(S, V_{x})(x)$.
(3): It is sufficient to verify
For any $S, S' \in \mathfrak{S}$, there exists $T \in \mathfrak{S}$ with $T \supset S, S'$. In which case, for any $U, U' \in \fU$,
\[E(T, U \cap U') \subset E(S \cup S', U \cap U') \subset E(S, U) \cap E(S', U')\]For any $U \in \fU$, $\Delta \subset U$. Thus the diagonal in $X^{T}$ is in $E(S, U)$ for any $S \in \mathfrak{S}$.
For any $U \in \fU$, there exists $V \in \fV$ with $V \circ V \subset U$. Thus for any $S \in \mathfrak{S}$,
\[E(S, V) \circ E(S, V) \subset E(S, V \circ V) \subset E(S, U)\]
By Proposition 5.1.8, $\mathfrak{E}$ is a fundamental system of entourages for the uniformity that it generates.$\square$