Fiber (mathematics)


In mathematics, the term fiber (US English) or fibre (British English) can have two meanings, depending on the context:

  1. In naive set theory, the fiber of the element in the set under a map is the inverse image of the singleton under [1]
  2. In algebraic geometry, the notion of a fiber of a morphism of schemes must be defined more carefully because, in general, not every point is closed.


Fiber in naive set theoryEdit

Let   be a function between sets.

The fiber of an element   (or fiber over  ) under the map   is the set

that is, the set of elements that get mapped to   by the function. It is the preimage of the singleton   (One usually takes   in the image of   to avoid   being the empty set.)

The collection of all fibers for the function   forms a partition of the domain   The fiber containing an element   is the set   For example, the fibers of the projection map   that sends   to   are the vertical lines, which form a partition of the plane.

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  

Fiber in algebraic geometryEdit

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  

Fibers in topologyEdit

Every fiber of a local homeomorphism is a discrete subspace of its domain. If   is a continuous function and if   (or more generally, if  ) is a T1 space then every fiber is a closed subset 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.

See alsoEdit


  1. ^ Lee 2011, p. 69, Above the Ex. 3.59.


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