A fiber of a map f:X->Y is the preimage of an element y in Y. That is,

 f^(-1)(y)={x in X:f(x)=y}.

For instance, let X and Y be the complex numbers C. When f(z)=z^2, every fiber consists of two points {z,-z}, except for the fiber over 0, which has one point. Note that a fiber may be the empty set.

In special cases, the fiber may be independent, in some sense, of the choice of y in Y. For instance, if f is a covering map, then the fibers are all discrete and have the same cardinal number. The example f(z)=z^2 is a covering map away from zero, i.e., f(z)=z^2 from the punctured plane C-{0} to itself has a fiber consisting of two points.

When pi:E->M is a fiber bundle, then every fiber is isomorphic, in whatever category is being used. For instance, when E is a real vector bundle of bundle rank k, every fiber is isomorphic to R^k.

See also

Bundle Rank, Complex Number, Covering Map, Fiber Bundle, Map, Whitney Sum

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Cite this as:

Weisstein, Eric W. "Fiber." From MathWorld--A Wolfram Web Resource.

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