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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 theory

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 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

Fibers in topology

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 also


  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.