In Hatcher's Algebraic Topology book it is noted after 4.61 that:

fiber preserving map + homotopy equivalence $\Rightarrow$ fiber homotopy equivalence.

## Question:

Could there be two fibrations over the **same** base space where the total spaces ** are** homotopy equivalent, but there is

**no**fiber homotopy equivalence between them? (and therefore also no fiber preserving map)

If so, I would be glad to have a simple example.