Exercise 19, § 2

Let $X$ be a Hausdorff uniform space and let $G$ be a group of homeomorphisms of $X$ onto itself.

a) Show that if $G$ is equicontinuous and discrete with respect to the topology of pointwise convergence, then $G$ is closed in $\mathscr{F}_s(X; X)$.

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

c) Let $X$ be the topological sum of two spaces $X_1, X_2$, each homeomorphic to $\mathbf{R}$, so that $X$ can be identified with the product space $\mathbf{R} \times \{1, 2\}$. Endow $X$ with the product uniformity. For each pair $(m, n)$ of rational integers let $u_{mn}$ denote the homeomorphism of $X$ defined by $u_{mn}(x_1) = x_1 + m\alpha + n\beta, \ u_{mn}(x_2) = x_2 + m\gamma + n\delta$, where $x_1 \in X_1$ and $x_2 \in X_2$, and $\alpha, \beta, \gamma, \delta$ are four non-zero real numbers such that $\alpha/\beta$ and $\gamma/\delta$ are irrational and distinct. Show that the $u_{mn}$ form a group of homeomorphisms of $X$ which is equicontinuous and discrete with respect to the topology of pointwise convergence, but that $G$ (with the discrete topology) is not proper at any point of $X$.