Proposition 6

Let $X$ be a topological space, $Y$ a uniform space and $\mathscr{G}$ a set of subsets of $X$. For each $A \in \mathscr{G}$ and each closed entourage $V$ of $Y$, the traces on $\mathscr{C}(X; Y) \times \mathscr{C}(X; Y)$ of $W(A, V)$ and $W(\overline{A}, V)$ are the same.

For if $u, v$ are continuous mappings of $X$ into $Y$, the mapping $x \mapsto (u(x), v(x))$ of $X$ into $Y \times Y$ is continuous, and the hypothesis that $(u(x), v(x)) \in V$ for all $x \in A$ therefore implies that $(u(x), v(x)) \in \overline{V} = V$ for all $x \in \overline{A}$ (Chapter I, § 2, no. 1, Theorem 1).

If $\bar{\mathscr{G}}$ denotes the set of closures in $X$ of the sets of $\mathscr{G}$, Proposition 6 shows that, on $\mathscr{C}(X; Y)$, the structures of $\mathscr{G}$-convergence and $\bar{\mathscr{G}}$-convergence are identical.

G-convergence in spaces of continuous mappings