Proposition 4

Let $X, Y$ be two sets, let $(X_\lambda)_{\lambda \in L}$ be a family of sets and let $(Y_\mu)_{\mu \in M}$ be a family of uniform spaces. For each $\lambda \in L$, let $\mathscr{S}_\lambda$ be a set of subsets of $X_\lambda$, let $g_\lambda$ be a mapping of $X_\lambda$ into $X$, and let $\mathscr{S}$ be the set of subsets of $X$ which is the union of the sets $g_\lambda(\mathscr{S}_\lambda)$. For each $\mu \in M$, let $f_\mu$ be a mapping of $Y$ into $Y_\mu$, and endow $Y$ with the coarsest uniformity for which the $f_\mu$ are uniformly continuous. Then the uniformity of $\mathscr{S}$-convergence on $\mathscr{F}(X; Y)$ is the coarsest uniformity which makes uniformly continuous the mappings $u \to f_\mu \circ u \circ g_\lambda$ of $\mathscr{F}(X; Y)$ into $\mathscr{F}_{\mathscr{S}_\lambda}(X_\lambda, Y_\mu)$.

These propositions are immediate consequences of the description of a fundamental system of entourages for the uniformity of $\mathscr{S}$-convergence given in no. 2, Remark 1; the details of the proofs are left to the reader. Proposition 4 implies, in particular: