I am interested in proving that a fibre bundle $p:E\to M$ of manifolds with contractible fibres is a homotopy equivalence. I know that this question has already been answered here or here by usaing a $CW-$structure on the spaces anche the Whitehead theorem, and these proofs are clear to me. However, I am also interested in a more "topological" proof. In particular, in a comment to this question the user "Tyrone" quotes the following theorem:
If there exists a numerable covering of $M$ for which each restriction $E|_U \to U $ is a homotopy equivalence over $U$, then $E \to M$ is a homotopy equivalence over $M$.
They gave the following reference: Theorem 7.57 in James's book General Topology and Homotopy Theory.
I found the book and tried looking at the proof, but, to be honest, the language used in the book is a bit obscure to me (expecially because this theorem is near the end, so they use a notation that is a bit hard for me to track back in the paragraphs where the definitions are given).
If any of you had a different proof of this fact, or even a different reference, I would thank you very much.
Alternatively, I would aprreciate any possible different solution to my original problem, which does not use a $CW-$structure.