Let $F \rightarrow E_i \rightarrow X_i$ be a bundle with fibre $F$ for i=1,2. Let $f:E_1 \rightarrow E_2$ be a bijective continuous map and $h: X_1 \rightarrow X_2$ a homeomorphism.

Is f then also a heomeomorphism? If not, what further properties are needed for f to be a hemeomorphism?