Fiber (mathematics)

Short description: Set of all points in a function's domain that all map to some single given point

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

f−1({y})={x:f(x)=y}

This set is often denoted as f−1(y), even though this notation is inappropriate since the inverse relation f−1 of f is not necessarily a function.

## Contents

* 1 Properties and applications
* 1.1 In naive set theory
* 1.2 In topology
* 1.3 In algebraic geometry
* 2 See also
* 3 References

## Properties and applications

### In naive set theory

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

{f−1(y):y∈Y}={{x∈X:f(x)=y}:y∈Y}

which is a partition of the domain set X. Note that y must be restricted to the image set Y of f, since otherwise f−1(y) would be the empty set which is not allowed in a partition. The fiber containing an element x∈X is the set f−1(f(x)).

For example, let f be the function from ℝ2 to ℝ that sends point (a,b) to a+b. The fiber of 5 under f are all the points on the straight line with equation a+b=5. The fibers of f are that line and all the straight lines parallel to it, which form a partition of the plane ℝ2.

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

If f is a real-valued function of several real variables, the fibers of the function are the level sets of f. If f is also a continuous function and y∈ℝ is in the image of f, the level set f−1(y) will typically be a curve in 2D, a surface in 3D, and, more generally, a hypersurface in the domain of f.

The fibers of f are the equivalence classes of the equivalence relation ≡f defined on the domain X such that x′≡fx″ if and only if f(x′)=f(x″).

### In topology

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

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

A function between topological spaces is called monotone if every fiber is a connected subspace of its domain. A function f:ℝ->ℝ 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 f between topological spaces X and Y whose fibers have certain special properties related to the topology of those spaces.

### In algebraic geometry

In algebraic geometry, if f:X->Y is a morphism of schemes, the fiber of a point p in Y is the fiber product of schemes X×YSpec⁡k(p) where k(p) is the residue field at p.

## See also

* Fibration
* Fiber bundle
* Fiber product
* Preimage theorem
* Zero set

## References

1. ↑ Lee, John M. (2011). Introduction to Topological Manifolds (2nd ed.). Springer Verlag. ISBN 978-1-4419-7940-7. https://www.springer.com/gp/book/9781441979391.

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