Exercise 20, § 2

Let $X$ be a locally compact metric space, $G$ a group of homeomorphisms of $X$, equicontinuous and discrete for the topology of simple convergence (equal to the topology of compact convergence on $G$).

(a) Now prove that if there exists a point $x \in X$ and a sequence $(u_n)$ of distinct elements of $G$ such that the sequence of the $u_n(x)$ is convergent, then there exists a sequence $(v_n)$ of distinct elements of $G$ such that $\lim_{n \to \infty} v_n(x) = x$. Deduce that there then exist two compact neighbourhoods K, L of x such that $v_n(K) \subset L$ for all n.

(b) The hypotheses on $G$ do not imply that $G$ is proper at at least one point of $X$ (Exercise 19 c)). We say that $G$ is K-discrete for a compact subset K of $X$, if $G$ is discrete for the topology of uniform convergence in K (or, equivalently, for the topology of simple convergence in K). Now prove that if $G$ is K-discrete for every compact subset K of $X$, $G$ operates properly in $X$ (use a)).

(c) Suppose that $G$ is uniformly equicontinuous and that $X$ is connected (therefore of countable type (IX, p. 95, Exercise 17)); now prove that $G$ operates properly in $X$. (Using X, p. 45, Exercise 13 d), show that if a sequence $(v_n)$ of distinct elements of $G$ were such that $\lim_{n \to \infty} v_n(x) = x$ for a $x \in X$, there would exist a subsequence of $(v_n)$ tending to the identity for the discrete topology). Give an example where X is connected but G is not K-discrete for every compact subset of X (take for X the union, in $\mathbf{R}^2$, of the segments $-1 \leq x \leq 1, y = 0$ and $x = 0, 0 \leq y \leq 1$).