This definition seems a bit weird. The definition should be of either smooth vector bundle or the general notion of vector bundle without restricting cases where the base space is a smooth manifold. The definition of smooth vector bundle includes restrictions on the transition functions too.
The phrasing is too dense: hard to edit from here. In what sense is the fibre 'isomorphic' to V? What category is that in?
Charles Matthews 09:15, 29 Nov 2003 (UTC)