Proposition 5

Let $\Phi$ be a set, $Y$ a uniform space and $\mathscr{S}$ a set of subsets of $X$. Then a filter $\Phi$ on $\mathscr{F}_\mathscr{S}(X; Y)$ converges to $u_0$ if and only if $\Phi$ is a Cauchy filter with respect to the uniformity of $\mathscr{S}$-convergence and converges pointwise to $u_0$ in $B = \bigcup_{A \in \mathscr{S}} A$.

Since the structure of pointwise convergence in $B$ is coarser than that of $\mathscr{S}$-convergence, it is enough to show that for each $A \in \mathscr{S}$ and each closed entourage $V$ of $Y$, $W(A, V)$ is closed in $B$ with respect to the topology of pointwise convergence (Chapter II, § 3, no. 3, Proposition 7). Now $W(A, V)$ is the intersection of the inverse images of $V$ under the mappings $(u, v) \to (u(x), v(x))$ as $x$ runs through $A$; these mappings are continuous with respect to the topology of pointwise convergence (no. 2, Remark 6), and the result follows.