Exercise 17, § 2

Let $X$ be a compact space and $\Gamma$ a countable group of homeomorphisms of $X$. Suppose that the orbit space $X/\Gamma$ is Hausdorff (and therefore compact).

a) Show that, for each $x \in X$, the orbit $\Gamma(x)$ of $x$ is finite (observe that if it were infinite it would have no isolated point, and use Baire's theorem (Chapter IX, § 5, no. 3, Theorem 1)).

b) For each $x_0 \in X$, let $\Delta(x_0)$ be the normal subgroup of $\Gamma$ consisting of homeomorphisms which leave fixed all the points of the orbit $\Gamma(x_0)$. Show that if $\Delta(x_0)$ is finitely generated, then $\Gamma$ is equicontinuous at $x_0$. [Let $f_i (1 \leq i \leq m)$ be the generators of $\Delta(x_0)$, and let $g_k (1 \leq k \leq n)$ be representatives of each of the cosets of $\Delta(x_0)$ in $\Gamma$ other than $\Delta(x_0)$ itself. Take a neighbourhood $V$ of $x_0$ such that none of the sets $f_i(V)$, $f_i^{-1}(V)$ meets any of the sets $g_k(V)$, and then use the fact that the equivalence relation defined by $\Gamma$ is closed (Chapter I, § 10, no. 4, Proposition 8).]

c) Without making any hypothesis on $\Delta(x_0)$, suppose that $x_0$ has a countable fundamental system of connected neighbourhoods in $X$. Show that $\Gamma$ is equicontinuous at $x_0$ [same method as in b)]. Deduce that if $X$ is locally connected, then $\Gamma$ is equicontinuous.

d) Take $X$ to be the compact subspace of $\mathbf{R}$ consisting of the points $0, 1, 1/n$ and $1 + 1/n$ ($n$ an integer $\geq 2$). Give an example of a countable group $\Gamma$ of homeomorphisms of $X$, such that $X/\Gamma$ is Hausdorff but $\Gamma$ is not equicontinuous.