Example 1

Let F be a topological space and let G be a topological group operating continuously on the left in F. If the topological space G is endowed with the operation of G by right translations, the space $G\times^GF$ is canonically identified with F in the following way. The continuous mappings $\varphi : F\rightarrow G\times F$ and $\psi : G\times F\rightarrow$ F defined by $\varphi (f) = (e, f)$ (where $e$ denotes the identity element of G) and $\psi (g, f) =g\cdot f$ induce continuous mappings $\overline{\varphi}: F\rightarrow G\times^GF$ and $\overline{\psi}: G\times^GF\rightarrow F$ which are reciprocal to one another.