Exercise 20, § 2

Let $G$ be a group of homeomorphisms of a Hausdorff uniform space $X$. Suppose that $G$ is equicontinuous and that $G$, endowed with the discrete topology, operates properly on $X$. Suppose, furthermore, that the set of points $x \in X$, which are not fixed points of any homeomorphism $u \in G$ other than the identity, is dense in $X$. Under these conditions, show that there is an open set $F$ in $X$ such that $F \cap U(F) = \varnothing$ for all $u \in G$ other than the identity, and such that the canonical image of $F$ in the orbit space $X/G$ is a dense open subset of $X/G$, homeomorphic to $F$ (use Exercise 16 and Zorn’s lemma).