Exercise 17, § 2

Let $X$ be a compact space, $\Gamma$ a countable group of homeomorphisms of $X$; suppose that the orbit space $X/\Gamma$ is separated (hence compact).

a) Now prove that the orbit $\Gamma(x)$ of every $x \in X$ is a finite set (notice that if this set were infinite, it would have no isolated point and use the Baire theorem (IX, p. 55, th. 1)).

b) For each point $x_0 \in X$, let $\Delta(x_0)$ be the distinguished subgroup of $\Gamma$ formed by the homeomorphisms leaving invariant all the points of the orbit $\Gamma(x_0)$. Now prove that if $\Delta(x_0)$ has a finite number of generators, the group $\Gamma$ is equicontinuous at the point $x_0$. (Let $f_i$ ($1 \leq i \leq m$) be the generators of $\Delta(x_0)$, $g_k$ ($1 \leq k \leq n$) representatives of each of the classes of the group $\Gamma$ modulo $\Delta(x_0)$ distinct from $\Delta(x_0)$; take a neighbourhood $V$ of $x_0$ such that none of the sets $f_i(V)$, $f_i^{-1}(V)$ meets one of the sets $g_k(V)$; then use the fact that the equivalence relation defined by $\Gamma$ is closed (I, p. 78, prop. 8)).

c) Without assumption on $\Delta(x_0)$, suppose that $x_0$ has a fundamental system of connected neighbourhoods in $X$; show that $\Gamma$ is equicontinuous at the point $x_0$ (method analogous to that of b)). Deduce that if $X$ is locally connected, $\Gamma$ is equicontinuous.

d) Take for $X$ the compact subspace of $\mathbf{R}$ formed by 0, 1 and the points $1/n$ and $1 + \frac{1}{n}$ ($n$ integer $\geq 2$); give an example of a countable group $\Gamma$ of homeomorphisms of $X$ such that $X/\Gamma$ is separated, and which is not equicontinuous.