TITLE: Is this an Accurate General description of Line Bundles?
QUESTION [4 upvotes]: As a newbie to vector bundles, it seems like all vector bundles I have  run into ( not that many, I admit) need only two charts to be trivialized; one of these
  charts will contain the "trouble point" (the point that prevents the top space from
  being a global product), and one chart that will not contain the trouble point ( I am thinking of the Möbius band over the circle here ).
It also seems like the trivializations ultimately consist of unwrapping the twist
  in the trouble point in the top( again, in here, in the case of the Möbius bundle, 
  I am thinking of the lift of (1,0) in S^1 ) , unwrapping , and flattenning.
Is this an at least informally correct generalization of vector bundles ( or, at
  least of plane bundles, i.e., 1-bundles? )
Thanks.

REPLY [0 votes]: @ elgeorges:
What I mean by a single problem point is that we only need to do a single
   vertical cut on the bundle to be able to "flatten it", i.e., make it be
   the union of a copy of two disjoint product spaces. I  don't know if my statement
   is too informal, or maybe even wrong, but the fact that we only need two charts
   for the bundle would seem to reinforce this claim that there is one main problem
   point. I am also referring to the fact that --please correct me if I am wrong--
   that all the fibers except for one are of R with one orientation, and one fiber
   only has orientation opposite that of all the others; it is only at the attaching
   parts of the fundamental domain that the bundle becomes non-trivial. If the map
   y-->1-y  (after assignining coordinates (0,0), (0,1),(1,0),(1,1) to the sides of
   the square were homotopic to the map y->y , the bundle would be trivial. Is this
   correct?
{"set_name": "stack_exchange", "score": 4, "question_id": 19594}