Fiber (mathematics)


In mathematics, the fiber (US English) or fibre (British English) of an element under a function is the preimage of the singleton set ,[1]: p.69  that is

As an example of abuse of notation, this set is often denoted as , which is technically incorrect since the inverse relation of is not necessarily a function.

Properties and applications edit

In naive set theory edit

If   and   are the domain and image of  , respectively, then the fibers of   are the sets in


which is a partition of the domain set  . Note that   must be restricted to the image set   of  , since otherwise   would be the empty set which is not allowed in a partition. The fiber containing an element   is the set  

For example, let   be the function from   to   that sends point   to  . The fiber of 5 under   are all the points on the straight line with equation  . The fibers of   are that line and all the straight lines parallel to it, which form a partition of the plane  .

More generally, if   is a linear map from some linear vector space   to some other linear space  , the fibers of   are affine subspaces of  , which are all the translated copies of the null space of  .

If   is a real-valued function of several real variables, the fibers of the function are the level sets of  . If   is also a continuous function and   is in the image of   the level set   will typically be a curve in 2D, a surface in 3D, and, more generally, a hypersurface in the domain of  

The fibers of   are the equivalence classes of the equivalence relation   defined on the domain   such that   if and only if  .

In topology edit

In point set topology, one generally considers functions from topological spaces to topological spaces.

If   is a continuous function and if   (or more generally, the image set  ) is a T1 space then every fiber is a closed subset of   In particular, if   is a local homeomorphism from   to  , each fiber of   is a discrete subspace of  .

A function between topological spaces is called monotone if every fiber is a connected subspace of its domain. A function   is monotone in this topological sense if and only if it is non-increasing or non-decreasing, which is the usual meaning of "monotone function" in real analysis.

A function between topological spaces is (sometimes) called a proper map if every fiber is a compact subspace of its domain. However, many authors use other non-equivalent competing definitions of "proper map" so it is advisable to always check how a particular author defines this term. A continuous closed surjective function whose fibers are all compact is called a perfect map.

A fiber bundle is a function   between topological spaces   and   whose fibers have certain special properties related to the topology of those spaces.

In algebraic geometry edit

In algebraic geometry, if   is a morphism of schemes, the fiber of a point   in   is the fiber product of schemes

where   is the residue field at  

See also edit

References edit

  1. ^ Lee, John M. (2011). Introduction to Topological Manifolds (2nd ed.). Springer Verlag. ISBN 978-1-4419-7940-7.