Exercise 19, § 2

Let X be a separated uniform space, G a group of homeomorphisms of X onto itself.

(a) Now prove that if G is equicontinuous and discrete for the topology of simple convergence, it is closed in the space $\mathscr{F}_s(X; X)$.

(b) Now prove that if G, endowed with the discrete topology, is proper at at least one point of X (exerc. 18), then it is closed in $\mathscr{F}_s(X; X)$ and the topology of simple convergence induces on G the discrete topology.

(c) Let X be a topological sum space of two spaces $X_1, X_2$ homeomorphic to $\mathbf{R}$, such that X can be identified with the product space $\mathbf{R} \times \{1, 2\}$; it is endowed with the product uniform structure. For every pair $(m, n)$ of rational integers, denote by $u_{mn}$ the homeomorphism of X defined by $u_{mn}(x_1) = x_1 + m\alpha + n\beta, u_{mn}(x_2) = x_2 + m\gamma + n\delta$ for $x_1 \in X_1$ and $x_2 \in X_2$, where $\alpha, \beta, \gamma, \delta$ are four real numbers $\neq 0$ such that $\alpha/\beta$ and $\gamma/\delta$ are irrational and distinct. Now prove that the $u_{mn}$ form a group G of homeomorphisms of X, which is equicontinuous and discrete for the topology of simple convergence, but that G (endowed with the discrete topology) is proper at no point of X.