Fiber

A fiber of a map  is the preimage of an element . That is,

For instance, let  and  be the complex numbers . When , every fiber consists of two points , 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 . For instance, if  is a covering map, then the fibers are all discrete and have the same cardinal number. The example  is a covering map away from zero, i.e.,  from the punctured plane  to itself has a fiber consisting of two points.

When  is a fiber bundle, then every fiber is isomorphic, in whatever category is being used. For instance, when  is a real vector bundle of bundle rank , every fiber is isomorphic to .