Definition 2

Let $X$ be a set, $Y$ a uniform space, $\mathfrak{S}$ a set of subsets of $X$. The uniformity of uniform convergence in the sets of $\mathfrak{S}$, or simply the uniformity of $\mathfrak{S}$-convergence, is the coarsest uniformity on $\mathscr{F}(X; Y)$ which makes uniformly continuous the restriction mappings $u \to u|A$ of $\mathscr{F}(X, Y)$ into the uniform spaces $\mathscr{F}_u(A; Y)$, where $A$ runs through $\mathfrak{S}$. The uniform space obtained by endowing $\mathscr{F}(X; Y)$ with the uniformity of $\mathfrak{S}$-convergence is denoted by $\mathscr{F}_{\mathfrak{S}}(X; Y)$.

The topology induced by the uniformity of $\mathfrak{S}$-convergence is called the topology of $\mathfrak{S}$-convergence; it is the coarsest for which all the mappings $u \to u|A$ of $\mathscr{F}(X; Y)$ into $\mathscr{F}_u(A; Y)$ ($A \in \mathfrak{S}$) are continuous (Chapter II, § 2, no. 3, Proposition 4, Corollary).

A filter $\Phi$ on $\mathscr{F}(X; Y)$ converges to $u_0$ with respect to the topology of $\mathfrak{S}$-convergence if and only if $u|A$ converges uniformly to $u_0|A$ with respect to $\Phi$ for all $A \in \mathfrak{S}$ (Chapter I, § 7, no. 6, Proposition 10), and $\Phi$ is therefore said to converge uniformly to $u_0$ on the sets of $\mathfrak{S}$.

A filter base $\Phi$ on $\mathscr{F}_{\mathfrak{S}}(X; Y)$ is a Cauchy filter base if and only if, for each $A \in \mathfrak{S}$, the image of $\Phi$ under the mapping $u \to u|A$ is a Cauchy filter base on $\mathscr{F}_u(A; Y)$ (Chapter II, § 3, no. 1, Proposition 4).

Let $f$ be a mapping of a topological (resp. uniform) space $Z$ into $\mathscr{F}_{\mathfrak{S}}(X; Y)$. Then $f$ is continuous (resp. uniformly continuous) if and only if, for each $A \in \mathfrak{S}$, the mapping $z \to f(z)|A$ of $Z$ into $\mathscr{F}_u(A; Y)$ is continuous (resp. uniformly continuous) (Chapter I, § 2, no. 3, Proposition 4; Chapter II, § 2, no. 3, Proposition 4).

Finally, let $M$ be a subset of $\mathscr{F}_{\mathfrak{S}}(X; Y)$; then $M$ is precompact if and only if, for each $A \in \mathfrak{S}$, the set of restrictions $u|A$ for $u \in M$ is a precompact subset of $\mathscr{F}_u(A; Y)$ (Chapter II, § 4, no. 2, Proposition 3).