Fiber
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المرجع الالكتروني للمعلوماتيه
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24-5-2021
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Fiber
A fiber of a map 
is the preimage of an element 
. That is,
![f^(-1)(y)=<span style=]() {x in X:f(x)=y}. " src="https://mathworld.wolfram.com/images/equations/Fiber/NumberedEquation1.gif" /> |
For instance, let 
and 
be the complex numbers 
. When 
, every fiber consists of two points ![<span style=]()
{z,-z}" src="https://mathworld.wolfram.com/images/equations/Fiber/Inline7.gif" />, 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 ![C-<span style=]()
{0}" src="https://mathworld.wolfram.com/images/equations/Fiber/Inline12.gif" /> 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 
.
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