Corollary 2

Let G be a topological group, let H be a subgroup of G, and let $p: G\rightarrow G/H$ be the canonical mapping. If the mapping $p$ possesses a continuous section over a nonempty open set of $G/H$, the mapping $p$ makes G into a principal fibre space with base $G/H$ and group H.

By left translations, every point of $G/H$ possesses a neighbourhood over which the mapping $p$ possesses a continuous section. On the other hand, for $(g, g')$ belonging to $G\times G$, put $\varphi (g, g') =$ $g^{-1}g'$. If $p(g) =p(g'),\varphi (g, g')$ belongs to H. The mapping $(g, g')\mapsto$ $(g, \varphi (g, g'))$ of $G\times_{G/H}G$ into $G\times H$ is continuous and is the inverse of the continuous mapping $\theta : G\times H\rightarrow G\times_{G/H}G$ defined by $\theta (g, h) = (g, gh)$, whence the corollary.