习题 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 $\tilde{\mathfrak{U}}_0$. 证明$X$的典范映射$x \to \{ \{ x \} \}$进入$\mathfrak{F}(\mathfrak{F}(X))$扩展到$\hat{X}$的同构到$\mathfrak{F}(\mathfrak{F}(X))$的封闭均匀子空间（带有均匀性$\mathfrak{U}_{00}$）（应用命题9 no. 4）。