Remark 3

The uniformity of $\mathscr{S}$-convergence is the inverse image, under the mapping $u \to (u|A)_{A \in \mathscr{S}}$ of $\mathscr{F}(X; Y)$ into $\prod_{A \in \mathscr{S}} \mathscr{F}_u(A; Y)$, of the uniformity of this product space (Chapter II, § 2, no. 6, Proposition 8). If $\mathscr{S}$ is a covering of $X$, this mapping is injective and $\mathscr{F}_\mathscr{S}(X; Y)$ is therefore isomorphic to the uniform subspace of $\prod_{A \in \mathscr{S}} \mathscr{F}_u(A; Y)$ which is the image of this mapping.