Proposition 8

Let G be a discrete group operating properly on a Hausdorff space X. Let x be a point of X and let K_x be the stabilizer of x. Then:

a) The subgroup K_x is finite and there is an open set U ⊂ X, containing x, which is stable under K_x, and on which the equivalence relation induced by the relation defined by G is the equivalence relation defined by K_x.

b) The canonical mapping U/K_x → X/G is a homeomorphism of U/K_x onto an open neighbourhood of the class of x in X/G.

By Proposition 7, K_x is finite. To construct an open set U which satisfies the required conditions, notice first that by Proposition 7 there is an open set U_0 containing x and such that the set K of s ∈ G for which s · U_0 ∩ U_0 ≠ ∅ is finite. Clearly K_x ⊂ K; let s_1, ..., s_n be the elements of K — K_x. If we put x_i = s_i · x (1 ≤ i ≤ n), then x_i ≠ x for each i; since X is Hausdorff, there is for each index i an open neighbourhood V_i of x and an open neighbourhood V'_i of s_i · x such that V_i ∩ V'_i = ∅. Let U_i = V_i ∩ s_i^{-1} · V'_i; then U_i is clearly open and contains x, and we have U_i ∩ s_i · U_i ⊂ V_i ∩ V'_i = ∅. Let U' = U_0 ∩ U_1 ∩ ... ∩ U_n; U' is open, contains x and is such that U' ∩ s · U' = ∅ for s ∉ K_x. Putting U = ⋂_{t ∈ K_x} t · U' we finally obtain an open set, stable under K_x, containing x and such that U ∩ s · U = ∅ for s ∉ K_x: U is the open set required.

The fact that the canonical mapping U/K_x → X/G is a homeomorphism of U/K_x onto an open set in X/G follows from Chapter I, § 5, no. 2, Proposition 4, since U is open and the equivalence relation defined by G is open (§ 2, no. 4, Lemma 2).