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