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

This set is often denoted as , even though this notation is inappropriate since the inverse relation of is not necessarily a function.

Properties and applications

In naive set theory

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

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

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


References

  1. Lee, John M. (2011). Introduction to Topological Manifolds (2nd ed.). Springer Verlag. ISBN 978-1-4419-7940-7.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.