Exercise 3, § 3

a) Let $\mathfrak{U}$ denote the additive uniformity on the real line $\mathbf{R}$, let $\mathfrak{U}_1$ denote the uniformity induced on $\mathbf{R}$ by that of the extended line $\overline{\mathbf{R}}$, and let $\mathfrak{U}_2$ denote the uniformity induced on $\mathbf{R}$ by that of the one-point compactification $\tilde{\mathbf{R}} = P_1(\mathbf{R})$ of $\mathbf{R}$. Show that $\mathfrak{U}$ is strictly finer than $\mathfrak{U}_1$, that $\mathfrak{U}_1$ is strictly finer than $\mathfrak{U}_2$, and that the topologies induced by these three uniformities are the same; also that the completions of $\mathbf{R}$ with respect to these three uniformities are respectively $\mathbf{R}$, $\overline{\mathbf{R}}$ and $\tilde{\mathbf{R}}$. The identity mapping of $\mathbf{R}$ extends by continuity to an injective but not surjective mapping $\mathbf{R} \to \overline{\mathbf{R}}$ and to a surjective but not injective mapping $\overline{\mathbf{R}} \to \tilde{\mathbf{R}}$.

b) Deduce from a) an example of two Hausdorff uniform spaces $X, Y$ and a uniformly continuous bijection $u : X \to Y$ whose extension by continuity $\bar{u} : \hat{X} \to \hat{Y}$ is neither injective nor surjective.*