Exercise 3, § 1

Let $X$ be a set, let $\mathscr{S}$ be a covering of $X$, let $Y$ be a non-Hausdorff uniform space and let $Y_0$ be the Hausdorff uniform space associated with $Y$ (Chapter II, § 3, no. 8). Show that the Hausdorff uniform space associated with $\mathscr{F}_{\mathscr{S}}(X; Y)$ is isomorphic to $\mathscr{F}_{\mathscr{S}}(X; Y_0)$.