Example 5

Let B be a topological space, let G be a topological group, and let E be a principal fibred B-space with group G. Let F be a topological homogeneous space relative to G (TG, III, p. 12). Let $y$ be a point of F and let $G_y$ be its stabiliser. The mapping $\varphi_y:x\mapsto (x, y)$ of E into $E\times F$ defines, by passing to quotients, a homeomorphism $\overline{\varphi}_y$ of $E/G_y$ onto $E\times^GF$. When the group G is abelian, the subgroup $G_y$ does not depend on the point $y$, but the homeomorphism $\overline{\varphi}_y$, in general, does depend on it.