12   QUOTIENT GROUPS       55
all p J> n and q *> n. Since J is right-invariant, this means that for all y e x_p
the intersection of x_q and the neighborhood V_n j; is not empty. We can then
define, by induction on «, a sequence (x_n) in G such that x_n e x_n and
x_n+1 g V_n^c_n for all «. Hence it follows, by induction on/?, that for allp > 0 we
have x_n+peV_n+p__lV_n+p__2-'V_n+1V_nx_nciV_n^x_n (because V*_+lcV_n for
all n). Hence the sequence (x_n) is a right Cauchy sequence in G. Q.E.D.
Let G be a topological group, and let E be a topological space on which G
acts continuously and transitively. Let x e E and let S^ be the stabilizer of x in
G; then (12.10.1) the continuous mapping h_x : si—>s • x of G into E, which by
hypothesis is surjective, factorizes canonically as
(12.11.4)	h_x: G 2+ G/$_x % E,
where n_x is the canonical mapping of G onto the homogeneous space G/S_X,
and/_x is the bijection s$_x\-+s • x.
It follows from (12.10.6) that f_x is a continuous bijection, but it is not
necessarily a homeomorphism (Section 12.12, Problem 2).
(12.11.5)	With the notation above, the bijection f_x : G/S_X -> E is a homeomor-
phism for each x e E if and only if there exists a point x_QeE such that the
mapping h_XQ: s \-> s • x_q transforms each neighborhood of e in G into a neigh-
borhood of x_0 in E.
In view of (12.10.6) we need only prove that the condition is sufficient.
Notice first that each xeE can by hypothesis be written in the form
x = t - x_0 for some t e G. If V is a neighborhood of e, it follows from the
hypothesis that V - x = (Vf) • x_0 is a neighborhood of x, because we can
write (Vf) • x_0 = f • ((f ~^JVf) • jc_q), and t ~^lVt is a neighborhood of e in G
(12.8.3), and y*-+s-y is a homeomorphism of E onto E, for each seG
(12.10.3). Since each open set in G/S_X is the image under n_x of some open set
in G, it is enough to show that for each open set U in G, and each x e E, the
set h_x(U) =f_x(n_x(U)) = U • x is open in E. Now, for each fell, the set
t ~^l\J is a neighborhood of e, and therefore (t ~^1U) - x is a neighborhood of x
in E, from the first part of the proof. Hence U • x = t - ((t ^-1U) • x) is a
neighborhood of t • x; by (3.6.4), the proof is complete.
12. QUOTIENT GROUPS
If H is a normal subgroup of a group G, then xH = Ex for each x e G and
therefore the sets G/H and H\G are identical; and G/H has a group structure