Exercise 18, § 3

Let $X$ be a locally compact space endowed with a uniformity compatible with its topology, and let $\Gamma$ be the group of homeomorphisms of $X$. For each entourage $V$ of $X$ and each compact subset $K$ of $X$, let $G(K, V)$ denote the set of pairs $(u, v)$ of homeomorphisms of $X$ such that $(u(x), v(x)) \in V$ and $(u^{-1}(x), v^{-1}(x)) \in V$ for all $x \in K$.

a) Show that the sets $G(K, V)$ form a fundamental system of entourages of a uniformity $\mathscr{U}$ on $\Gamma$ and that the topology induced by this uniformity is the topology $\mathscr{T}_\beta$ defined in no. 5.

b) Show that if $X$ is complete with respect to its uniformity, then $\Gamma$ is complete with respect to the uniformity $\mathscr{U}$.

c) Show that $\Gamma$ is complete with respect to the two-sided uniformity defined by the topology $\mathscr{T}_\beta$ [use Exercise 16 c)].

d) Take $X$ to be the locally compact subspace of $\mathbf{R}$ consisting of points of the form $n + 2^{-m}$ ($n \in \mathbf{Z}$, $m$ an integer $\geqslant 1$). Show that on the group $\Gamma$ neither the uniformity $\mathscr{U}$ nor the uniformity of compact convergence is comparable with any of the three uniformities (left, right and two-sided) defined by the topology $\mathscr{T}_\beta$.