Remark

Let $X$ be a topological space, $Y$ a uniform space and $\mathscr{S}$ a set of subsets of $X$. Then on every precompact subset $H$ of $\mathscr{F}_{\mathscr{S}}(X; Y)$, the uniformity of $\mathscr{S}$-convergence is the same as the uniformity of pointwise convergence in $B = \bigcup_{A \in \mathscr{S}} A$. We can reduce to the case where $B = X$ and $Y$ is Hausdorff and complete; for if $j$ is the canonical injection $B \to X$ and $i$ the canonical mapping $Y \to \hat{Y}$, the uniformity of $\mathscr{S}$-convergence on $\mathscr{F}(X; Y)$ is the inverse image of the uniformity of $\mathscr{S}$-convergence on $\mathscr{F}(B; \hat{Y})$ under the mapping $\theta : u \to i \circ u \circ j$ (§ 1, no. 4, Proposition 4), and $H$ is precompact if and only if $\theta(H)$ is (Chapter II, § 4, no. 2, Proposition 3). This being so, if $B = X$ and $Y$ is Hausdorff and complete, $\mathscr{F}_{\mathscr{S}}(X; Y)$ is Hausdorff and complete (§ 1, no. 2, Proposition 1 and no. 5, Theorem 1); hence the closure $\overline{H}$ of $H$ in this space is compact. On $\overline{H}$, the topology of pointwise convergence is Hausdorff (§ 1, no. 2, Proposition 1) and coarser than that of $\mathscr{S}$-convergence; hence these two topologies coincide (Chapter I, § 9, no. 4, Theorem 2, Corollary 3) and consequently so do the uniformities of $\mathscr{S}$-convergence and pointwise convergence (Chapter II, § 4, no. 1, Theorem 1).