Exercise 7, § 3

Let $X$ be a Hausdorff uniform space, $\mathfrak{U}$ its uniformity, $\mathfrak{U}_0$ the uniformity induced on $\mathfrak{F}(X)$ by the uniformity $\tilde{\mathfrak{U}}$ ($§ 2$, Exercise 6 $b$), $\mathfrak{U}_{00}$ the uniformity induced on $\mathfrak{F}(\mathfrak{F}(X))$ by the uniformity $\tilde{\mathfrak{U}}_0$. Show that the canonical mapping $x \to \{ \{ x \} \}$ of $X$ into $\mathfrak{F}(\mathfrak{F}(X))$ extends to an isomorphism of $\hat{X}$ onto a closed uniform subspace of $\mathfrak{F}(\mathfrak{F}(X))$ (carrying the uniformity $\mathfrak{U}_{00}$) (use Proposition 9 of no. 4).