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 
.
