Proposition 10

Let X be a nonempty connected topological space, let G be a discrete group operating continuously on the left in X, and let M be a subset of X such that $G\cdot M = X$. One of the two following hypotheses is made:

(i) The set M is open;

(ii) The set M is closed and the covering $(g\cdot M)_{g\in G}$ of X is locally finite.

Let S be the set of elements $g\in G$ such that $M\cap g\cdot M=\not\emptyset$, let T be the set of pairs $(s, t)\in S\times S$ such that $M\cap s\cdot M\cap st\cdot M=\not\emptyset$. Let F be the group $F(S,\mathbf{r})$ defined by the generator set S and by the set $\mathbf{r}$ of relators $str^{-1}$ for $r, s, t\in$ S such that $(s, t)\in$ T and $r$ = $st$; for $s\in S$, let $x_s$ denote the image of $s$ under the canonical mapping of S into F. There exists a unique group homomorphism $\varphi : F\rightarrow G$ such that $\varphi (x_s) =s$ for every $s\in S$. It is surjective; it is an isomorphism if the space X is simply connected, or, more generally, if every covering of X which is trivializable over M is trivializable.

The homomorphism $\varphi$ is surjective (I, p. 133, lemma 3). With the notation of this No.$^o$, the covering Y of X is trivializable, since it is trivializable over M. By lemma 5, Y is connected. Consequently, $p: Y\rightarrow X$ is a homeomorphism and the group N is reduced to the identity element. Consequently, the homomorphism $\varphi : F\rightarrow G$ is a group isomorphism.