41. Vector bundles are special fiber bundles , those whose fibers are vector spaces and whose cocycle respects the vector space structure. 42. Examples of non-trivial fiber bundles include the M�bius strip and Klein bottle, as well as nontrivial covering spaces. 43. Since the projections of fiber bundles and covering maps are locally natural projections of products, these are also open maps. 44. Proximity focused second generation tubes could also be inverted by using a fiber bundle with a 180 degree twist in it. 45. More generally still, we can consider frame bundles associated to any vector bundle, or even arbitrary principal fiber bundles . 46. Anatomically, this would make sense since the cerebellum is connected to the circuit with many fine fibers and fiber bundles . 47. A principal-bundle, where denotes any topological group, is a fiber bundle together with a Hausdorff and possibly paracompact. 48. In that example, the base space of a fiber bundle is pulled back, in the sense of precomposition, above. 49. The mysium damage disrupts calcium homeostasis in injured fibers and fiber bundles , resulting in necrosis that peaks about 48 hours after exercise. 50. In algebraic topology, these arise particularly in fiber bundles , where they yield maps that have the opposite of the usual variance.
