Definition of a Fiber Bundle
I do not especially like commutation diagrams and I say symbolically the same thing as the Wikipedia definition:
A fiber bundle is (E, B, π, F) where:
- E, B and F are topological spaces,
- π is a map from E onto B,
- For all x in E there is an open neighborhood U (⊂B) of π(x) and a homeomorphism φ from U×F to π−1(U) (⊂E) such that for all b∊U and f∊F π(φ(b, f)) = b.
My φ is the inverse of the φ of the article.
Auxilliary Definition of Atlas
A chart is a map from some non empty open set in B into E.
Question:
Is an atlas necessary, not as part of the ‘data’ but as something that must exist in order to be a fiber bundle?
Intuitively it seems to me that an atlas could be constructed for any fiber bundle such as defined above.
Given an atlas, the above seems to be satisfied; for any x in E x will be in some chart of the atlas and
I see no ready definition of equivalence between fiber bundles.