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? 



When is a bijective map between bundles a homeomorphism?
It has been pointed out in the comments that this sort of thing cannot hold for arbitrary fiber bundles.


To follow up on euklid345's comment regarding vector bundles, there is a statement of this type for arbitrary principal bundles, assuming the appropriate definitions. There's a detailed discussion of this in my course notes, available at <http://sofia.nmsu.edu/~ramras/601.html> (see p. 15 of Lectures 3-5). I believe it's also somewhere in Husemoller's book.


