Exercise 6, § 1

a) Let $X$ be a completely regular space and let $\mathscr{U}$ be a uniformity compatible with the topology of $X$. Let $\mathscr{U}^*$ be the coarsest uniformity on $X$ which makes uniformly continuous all the mappings of $X$ into $[0, 1]$ which are uniformly continuous with respect to $\mathscr{U}$. Show that the uniformity $\mathscr{U}^*$ is Hausdorff and compatible with the topology of $X$, and that $X$ is precompact with respect to $\mathscr{U}^*$.

b) Let $K$ be a compact space. Show that every mapping $f : X \to K$ which is uniformly continuous with respect to $\mathscr{U}$ is also uniformly continuous with respect to $\mathscr{U}^*$ (note that the unique uniformity on $K$ is the coarsest for which all the continuous mappings of $K$ into the interval $[0, 1]$ are uniformly continuous). Hence show that $\mathscr{U}^*$ is the finest of the uniformities on $X$ which are coarser than $\mathscr{U}$ and with respect to which $X$ is precompact.