Exercise 16, § 3

Let $X$ be a uniform space and let $\Gamma$ be the group of all automorphisms of the uniform structure of $X$.

a) Show that the topology of uniform convergence on $\Gamma$ is compatible with its group structure (use Exercises 12 and 15). Moreover, the uniformity induced on $\Gamma$ by the uniformity of uniform convergence in $X$ coincides with the right uniformity of the topological group $\Gamma$.

b) If $X$ is the compact interval $[0, 1]$ of $\mathbf{R}$, show that the topological group $\Gamma$ defined in a) has no completion [construct a sequence of homeomorphisms $(u_n)$ of $X$ which is uniformly convergent but such that the sequence $(u_n^{-1})$ is not uniformly convergent].

c) Show that if $X$ is a complete Hausdorff uniform space, then the topological group $\Gamma$ is complete with respect to its two-sided uniformity (Chapter III, § 3, Exercise 6). [Observe that if $\Phi$ is a Cauchy filter on $\Gamma$ with respect to this uniformity, then $\Phi(x)$ and $\Phi^{-1}(x)$ converge in $X$ for all $x \in X$.]