Exercise 16, § 2

Let $X$ be a Hausdorff uniform space and let $\Gamma$ be an equicontinuous group of homeomorphisms of $X$. The equivalence relation $R$ defined by $\Gamma$ on $X$ is open (Chapter I, § 5, no. 2).

a) Show that if every orbit of $\Gamma$ is closed in $X$, then the orbit space $X/\Gamma$ is Hausdorff.

b) Show that if every orbit of $\Gamma$ is compact, then the relation $R$ is closed (use Chapter I, § 5, no. 4, Proposition 10).

c) Give an example where $X$ is compact but none of the orbits of $\Gamma$ is closed in $X$ (cf. Chapter III, § 2, Exercise 29).