I want to model certain physical concept using fiber bundles, because I believe it is the most suitable language therefor. I know what both the base space and the fiber in this situation are. Do these determine the bundle uniquely (like when a structure group is involved)? I mean, is it necessary to specify a total space? Can it somehow (topologically/geometrically speaking) be described as the disjoint union of fibers?
I must add that the base space and the fibers are smooth manifolds, deprived a priori of any further structure.
Thanks in advance for your attention.