Let $\eta_i:E_i\rightarrow B_i$ be fiber bundles, and let $f_i:\eta_i\rightarrow\eta_{i+1}$ be maps of fiber bundles.
Assuming that the bundles maps $\eta_i$ are maps of smooth compact manifolds, and that the maps $f_i$ are inclusions of regular submanifolds, is the union $\bigcup_i\eta_i$ of the fiber bundles a fiber bundle?
I'm struggling to find a trivializing open cover of the base space of $\bigcup_i\eta_i$, but I think that it should be possible.
