Example 3

Analogously, let E be a topological space and let G be a topological group operating continuously on the right in E. If the topological space G is endowed with the operation of G by left translations, the space $E\times^GG$ is identified with E.

Let E and F be topological spaces and let G be a topological group operating continuously on the right in E and on the left in F. Let B be a topological space and let $p: E\rightarrow B$ be a continuous mapping such that $p(x\cdot g) =p(x)$ for $x\in E$ and $g\in G$. The mapping $p\circ$ pr$_1: E\times F\rightarrow B$ defines, by passing to the quotient, a continuous mapping $p^F: E\times^GF\rightarrow B$ and the canonical mapping $\pi : E\times F\rightarrow E\times^GF$ is a B-morphism.

Let $B'$ be a topological space and let $h: B'\rightarrow B$ be a continuous mapping. The group G operates continuously on the right in $B'\times_BE$ by the operation law $((b', x), g)\mapsto (b', x\cdot g)$. By composing the canonical isomorphism $((b', x), y)\mapsto (b',(x, y))$ of $(B'\times_BE)\times F$ onto $B'\times_B(E\times F)$ and the mapping Id$_{B'}\times \pi$ of $B'\times_B(E\times F)$ into $B'\times_B(E\times^GF)$, one obtains a continuous mapping $\lambda_0: (B'\times_BE)\times F\rightarrow B'\times_B(E\times^GF)$. It defines, by passing to the quotient, a continuous mapping