Let $Y$ be a compact manifold, $X$ a topological space and $f: X \to Y$ a surjective map. Suppose further that every point in $Y$ has arbitrarily small open neighbourhoods such that their preimages in $X$ are contractible.
Under these conditions, can we find a good cover for $Y$ such that its preimage is also a good cover for $X$ (i.e. such that any finite intersections of open sets in the cover is contractible)?
This would follow easily if the preimage of a contractible subspace in $Y$ was still contractible, but I have been unable to prove this or to find a counterexample.