Exercise 21, § 2

Let G be a group of homeomorphisms of a separated uniform space X. Suppose that G is equicontinuous, and that, endowed with the discrete topology, G operates properly in X. Suppose in addition that the set of the $x \in X$ which are not invariant under any homeomorphism $u \in G$ distinct from the identity, is everywhere dense in X. Under these conditions, show that there exists an open set F in X such that $F \cap u(F) = \varnothing$ for every homeomorphism $u \in G$ distinct from the identity, and that the canonical image of F in the space of orbits E/G is an open subset of E/G, everywhere dense and homeomorphic to F (use X, p. 45, Exercise 16 and Zorn's theorem).