Lemma 1

Let X be a topological space, let G be a discrete group operating properly in X, and let $p$ be the canonical projection of X onto $X/G$.

Let $x$ be a point of X, let $K_x$ be its stabilizer in G. Let $U_1$ be a neighbourhood of $x$ in X; there exists a neighbourhood U of $x$ in X, stable under $K_x$ and contained in $U_1$ such that $g\cdot U\cap U =\emptyset$ if $g\notin K_x$ and such that the canonical mapping $p$ induces a homeomorphism of $U/K_x$ onto a neighbourhood V of $p(x)$ in $X/G$.

By prop. 8 of TG, III, p. 32, there exists a neighbourhood $U_2$ of $x$ in X, stable under $K_x$, such that $g\cdot U_2\cap U_2=\emptyset$ if $g\notin K_x$ and such that the canonical mapping $p$ induces a homeomorphism of $U_2/K_x$ onto a neighbourhood of $p(x)$ in $X/G$.

Since the canonical mapping $U_2\rightarrow U_2/K_x$ is closed (TG, III, p. 28, prop. 2), there exists an open neighbourhood U of $x$ contained in $U_1\cap U_2$ which is stable under $K_x($I, p. 75, lemma). The equivalence relation in $U_2$ defined by $K_x$ is also open (TG, III, p. 10, lemma 2). It then follows from TG, I, p. 32, prop. 4, that the canonical mapping induces a homeomorphism of $U/K_x$ onto an open neighbourhood of $p(x)$ in $X/G$, whence the lemma.