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Clarifications:

Vector spaces are usually over R or C.
Vector bundles need to be 'locally trivial', whatever that means.
Whatever that means has a meaning! It means that the fibres are mapped "nicely" without any "twist". For instance let E be the total space, B the base space and F the fiber. When there is "Trivial fiber bundle", π: E = BxF ---> B happens, requiring at least surjectivity.

Some counter examples:

Mobius Strip
Klein Bottle

Vector Bundle: It's an attempt to form a precise definition of a family of vector spaces 'parameterized' by another space (topological, manifold or a variety) for every x in X there is a V(x) in VB