Remark 6

It follows immediately from Definitions 1 and 2 that, for every $x \in \bigcup_{A \in \mathscr{G}} A$, the mapping $u \to u(x)$ of $\mathscr{F}_\mathscr{G}(X; Y)$ into $Y$ is uniformly continuous. Hence, in particular, if $\overline{H}$ denotes the closure of a subset $H$ of $\mathscr{F}_\mathscr{G}(X; Y)$, we have $\overline{H}(x) \subset \overline{H(x)}$ for all $x \in \bigcup_{A \in \mathscr{G}} A$ (Chapter I, § 2, no. 1, Theorem 1).