- A fibration $\pi : E \to B$ over a contractible space is fiber-homotopy equivalent to the trivial fibration $B \times \pi^{-1}(b)$ for any point $b \in B$.
- A fiber bundle over a paracompact space is a fibration.
- A fiber bundle over a paracompact contractible space is trivial: the total space is homeomorphic to the product of the fiber and the base.
Question: If $\pi:E \to B$ is a fibration over a paracompact contractible space, must $E$ be homeomorphic to $B \times \pi^{-1}(b)$? (As opposed to just being homotopy equivalent, which doesn't need paracompactness.)