Exercise 18, § 2

Let $G$ be a separated topological group operating continuously in a separated topological space $X$; one says that $G$ is proper at a point $x_0 \in X$ if the orbit $G.x_0$ is closed in $X$ and if $G$ operates properly in $G.x_0$ (III, p. 27).

a) Let $G$ be a locally compact topological group operating continuously in a separated uniform space $X$. Suppose that the set of homeomorphisms $x \mapsto s.x$ of $X$ for $s \in G$ is equicontinuous. Now prove that if $G$ is proper at a point $x_0 \in X$, there exists a neighbourhood $V$ of $x_0$ such that the set of the $s \in G$ for which $s.V \cap V \neq \varnothing$ is relatively compact in $G$ (cf. III, p. 31, prop. 7). Deduce that the set $D$ of the points of $X$ where $G$ is proper is open in $X$ and that $G$ operates properly in $D$. If moreover $X$ is locally compact, now prove that $D$ is at the same time open and closed in $X$ (use X, p. 12, cor. 2).

b) In the numerical plane $\mathbf{R}^2$, let $E$ be the set formed by the origin $(0, 0)$ and the points $(0, 2^{-n})$ for $n$ integer $\geq 0$. For every integer $n \in \mathbf{Z}$ different from 0, denote by $u_n$ the restriction to $E$ of an affine linear mapping of $\mathbf{R}^2$ into itself such that $u_n(0, 0) = (n, 0)$ and $u_n(0, 2^{-|n|}) = (0, \theta^n)$, where $\theta$ is an irrational number such that $0 < \theta < 1$; let $X$ be the subspace (locally compact) of $\mathbf{R}^2$ union of $E$ and the $u_n(E)$ for $n \in \mathbf{Z}, n \neq 0$. We denote in addition by $u_0$ the identical map of $E$ onto itself, and define $u_n$ ($n \in \mathbf{Z}$) in the whole space $X$ by putting $u_n(u_m(x)) = u_{n+m}(x)$ for all $x \in E$ and all $m \in \mathbf{Z}$; the set $G$ of the $u_n$ is a group of homeomorphisms of $X$, which is provided with the discrete topology, so that $G$ operates continuously in $X$. Now prove that $G$ is proper at every point of $X$, but is not equicontinuous and does not operate properly in $X$.

c) Let $X$ be the subspace (not locally compact) of $\mathbf{R}^2$ (identified with $\mathbf{C}$), union of the half-plane $y > 0$ and the origin. Define a homeomorphism $u$ of $X$ onto itself by putting $u(0, 0) = (0, 0)$, and $u(re^{i\omega}) = re^{i\omega'}$, with $\omega' = \omega/2$ for $0 < \omega \leq \pi/2$, $\omega' = \omega - \pi/4$ for $\pi/2 \leq \omega \leq 3\pi/4$, $\omega' = 2\omega - \pi$ for $3\pi/4 \leq \omega < \pi$ ($r > 0, 0 < \omega < \pi$). Let $G$ be the monogenic group of homeomorphisms of $X$ generated by $u$, which is provided with the discrete topology. Now prove that $G$ is equicontinuous in $X$ and that the set of points of $X$ where $G$ is proper is the half-plane $y > 0$.

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