I'd like to understand Proposition $4.1$ p.$44$ of Husemoller "Fibre Bundles" which I'm currently studying.
The proof is the following :
What I don't understand is the conclusion, in particular the continuity of $\tau_1$. I think the last sentence should give the continuity of $\tau_1$ but I don't see how, given an open $W \subset G$, could I write $\tau_1^{-1}(W)$. Is the latter homeomorphic to $B \times B \times \tau^{-1}(W)$ which is open in the product?
Any help clarifying this detail would be appreciated.
Note $X^* \subset X\times X,X_1^* \subset(B_1\times X)\times(B_1\times X)$. We can think of the map $\tau_1$ as a composition of maps:
$$ X_1^* \xrightarrow{r} X^* \xrightarrow{\tau} G$$
where $r$ is simply the projection.