纖維叢概念 假設空間
是空間
和
的拓撲乘積。設
﹕=×→為向第一個乘積因子的投影映射﹐則對於任意x
﹐-1(x)均同胚於。因此可看作被分解為一族“纖維”{-1(x)}的聯合體。這些“纖維”相互聯合的方式是按照已知的乘積拓撲實現的
Möbius strip
Perhaps the simplest example of a nontrivial bundle E is the Möbius strip. It has the circle that runs lengthwise along the center of the strip as a base B and a line segment for the fiber F, so the Möbius strip is a bundle of the line segment over the circle. A neighborhood U of a point x ∈ B is an arc; in the picture, this is the length of one of the squares. The preimage π − 1(U) in the picture is a (somewhat twisted) slice of the strip four squares wide and one long. The homeomorphism φ maps the preimage of U to a slice of a cylinder: curved, but not twisted.
The corresponding trivial bundle B × F would be a cylinder, but the Möbius strip has an overall "twist". Note that this twist is visible only globally; locally the Möbius strip and the cylinder are identical (making a single vertical cut in either gives the same space).
