Example 2

Let $(E, p)$ be a principal fibre space with base B and group G; let $(E', p')$ be a principal fibre space with base $B'$ and group $G'$. We define a law of operation on the right of the group $G\times G'$ on the space $E\times E'$ by putting $(x, x')\cdot (g, g') = (x\cdot g, x'\cdot g')$ for $x\in E,x'\in E',g\in G$ and $g'\in G'$; this operation is continuous. Let U be an open subset of B and $f: U\times G\rightarrow \overset{-1}{p}(U)$ a trivialization of the U-space $(\overset{-1}{p}(U), p_U)$;

let $U'$ be an open subset of $B'$ and $f': U'\times G'\rightarrow (^-{p'}^1)(U')$ a trivialization of the $U'$-space ($(^-{p'}^1)(U'), p'_U$). The mapping $((b, b'),(g, g'))\mapsto$ $(f(b, g), f'(b', g'))$ is a $(U\times U')$-isomorphism $f''$ of $(U\times U')\times (G\times G')$

onto $\overset{-1}{p}(U)\times (^-{p'}^1)(U')$ and one has

It follows that the $(B\times B')$-space $E\times E'$, endowed with the law of operation of $G\times G'$ defined above, is a principal fibre space, called the product of the principal fibre spaces E and $E'$.