Example 5

Let B be a topological space and $(E, p)$ a principal fibre space with base B and group G. Let $B'$ be a topological space and $f: B'\rightarrow B$ a continuous mapping. The group G operates on the right in the fibre product $B'\times_BE$ by the law $(b', x)\cdot g= (b', x\cdot g)$, for $b'\in B',x\in E$ and $g\in G$. This operation is continuous and free; the orbits are the fibres of the mapping pr$_1: B'\times_BE\rightarrow B'$. It follows from Examples 1 and 4 above and Remark 2 of I, p. 16 that the $B'$-space $B'\times_BE$ is a principal fibre space with group G. It is called the principal fibre space deduced from $(E, p)$ by the base change $f$, or again the inverse image of $(E, p)$ by $f$.