Exercise 19, § 3

Let $H$ be an equicontinuous group of homeomorphisms of a uniform space $X$; endowed with the topology of pointwise convergence, $H$ is a topological group (no. 5, Corollary to Proposition 10).

a) Show that the left uniformity on $H$ is finer than the uniformity of pointwise convergence and that these two uniformities coincide if $H$ is uniformly equicontinuous.

b) Let $u$ be the homeomorphism of the real line $\mathbf{R}$ defined by $u(x) = x + 1$ for $x \leqslant 0$, $u(x) = x + 1/(x + 1)$ for $x \geqslant 0$. Let $H$ be the subgroup generated by $u$ in the group of all homeomorphisms of $\mathbf{R}$. Show that $H$ is equicontinuous and is discrete with respect to the topology of pointwise convergence, but that the uniformity of pointwise convergence on $H$ is not the discrete uniformity (observe that the limit $u^{n+1}(x) - u^n(x)$ is 0 as $n$ tends to $+\infty$).

c) Take $X$ to be the discrete space $\mathbf{N}$ of natural integers, endowed with the metric $d$ such that $d(m, n) = 1$ whenever $m \neq n$, and take $H$ to be the group of isometries of $\mathbf{N}$, which is uniformly equicontinuous. Show that the topological group obtained by endowing $H$ with the topology of pointwise convergence has no completion [same method as in Exercise 16 b)].

d) Suppose that $X$ is Hausdorff and complete and that $H$ is uniformly equicontinuous. Show that the Cauchy filters on $H$ with respect to the two-sided uniformity of the topological group $H$ converge in the space $\mathscr{F}_s(X; X)$; that the set $H'$ of their limit points is a uniformly equicontinuous group of homeomorphisms of $X$, which (endowed with the topology of pointwise convergence) is complete with respect to its two-sided uniformity; and that $H$ is dense in $H'$.