Example 6

Let B be a topological space, let G be a topological group, let $(E, p)$ be a principal fibred B-space with group G. Let H be a topological group and let $f: G\rightarrow H$ be a morphism of topological groups; let us endow H with the left operation of G given by $g\cdot h=f(g)h$.

Let $q: E\times H\rightarrow E\times^GH$ be the canonical mapping; it is open, hence universally strict (I, p. 20, corollary 11). The mapping $m: (x, h, h')\mapsto q(x, hh')$ of $E\times H\times H$ into $E\times^GH$ is continuous. Let $(x, h)\in E\times H,h'\in H$ and $g\in G$; we have $m(xg, f(g)^{-1}h, h') =$ $q(xg, f(g)^{-1}hh') =q(x, hh') =m(x, h, h')$. Consequently, there exists a unique mapping

such that $m'(q(x, h), h') =q(x, h, h')$ for all $x\in E$ and all $h, h'\in H$. Since $q$ is universally strict, the mapping $m'$ is continuous. This is a right operation of the topological group H on the B-space $E\times^GH$.

Let U be an open subset of B such that $E_U$ is isomorphic to the principal fibre bundle over U, $U\times G$. Endowed with the operation of H, the U-space $(E\times^GH)_U$ is identified with the principal fibre bundle $U\times H$. This proves that $E\times^GH$ is a principal fibre bundle over B with group H.