While group theory enables a notion of an equivalence class of groups, there is no common definition of equivalence between fiber bundles and thus no notion of an abstract fiber bundle. One of the requirements of a definition of the group of a bundle would supposedly require that equivalent fiber bundles produce isomorphic groups; but we can’t even state that requirement!

I paraphrase part of the Wikipedia definition:

- We require that for any point of the base space there is a neighborhood such that the product space of the neighborhood and fiber space is homeomorphic to [the inverse image of the neighborhood].

If the specs for a particular fiber bundle must include specs for the homomorphisms then we have lost the desirable property that many different sets of neighborhoods may apply to describing a given bundle. This would be ameliorated with an equivalence relation between two sets of neighborhoods and associated homomorphisms. It is such hair that Steenrod introduces to explain coordinate bundles.

Thus the ‘examples’ such as the Möbeus strip given in about every development don’t really fully specify a fiber bundle. Relying on this definition and taking notation from there, I attempt the simplest specification of the Möbeus fiber bundle. We use the reals [0, 1] as coordinates on the circle, identifying 0 and 1.

- The base space is the circle, B = [0, 1].

The fiber space is [0, 1] ⊂ ℝ

Two neighborhoods cover the base: {(.8, .2), (.1, .9)}.

The 1st homomorphism is (x, y) → (x, y) and the 2nd is