Definition 4

Let B be a topological space and let G be a topological group. A locally trivial fibre space X over B is said to be associated with a principal fibre bundle E over B with group G if there exists a topological space F on which the group G operates continuously on the left and a B-isomorphism of $E\times^GF$ onto X.

Let E be a principal fibre bundle over B with group G and let X be a locally trivial fibre space over B associated with E. If the principal fibre bundle E is trivializable, X is trivializable (Prop. 7). If $B'$ is a topological space and $h: B'\rightarrow B$ a continuous mapping, the locally trivial fibre space over $B'$, $B'\times_BX$, deduced from X by base change is associated with the principal fibre bundle over $B'$, $B'\times_BE$ (I, p. 105, lemma).