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 [math]\displaystyle{ y }[/math] under a function [math]\displaystyle{ f }[/math] is the preimage of the singleton set [math]\displaystyle{ \{ y \} }[/math],^[1]^:p.69 that is

[math]\displaystyle{ f^{-1}(\{y\}) = \{ x \mathrel{:} f(x) = y \} }[/math]

This set is often denoted as [math]\displaystyle{ f^{-1}(y) }[/math], even though this notation is inappropriate since the inverse relation [math]\displaystyle{ f^{-1} }[/math] of [math]\displaystyle{ f }[/math] is not necessarily a function.

Properties and applications

In naive set theory

If [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are the domain and image of [math]\displaystyle{ f }[/math], respectively, then the fibers of [math]\displaystyle{ f }[/math] are the sets in

[math]\displaystyle{ \left\{ f^{-1}(y) \mathrel{:} y \in Y \right\}\quad=\quad \left\{\left\{ x\in X \mathrel{:} f(x) = y \right\} \mathrel{:} y \in Y\right\} }[/math]

which is a partition of the domain set [math]\displaystyle{ X }[/math]. Note that [math]\displaystyle{ y }[/math] must be restricted to the image set [math]\displaystyle{ Y }[/math] of [math]\displaystyle{ f }[/math], since otherwise [math]\displaystyle{ f^{-1}(y) }[/math] would be the empty set which is not allowed in a partition. The fiber containing an element [math]\displaystyle{ x\in X }[/math] is the set [math]\displaystyle{ f^{-1}(f(x)). }[/math]

For example, let [math]\displaystyle{ f }[/math] be the function from [math]\displaystyle{ \R^2 }[/math] to [math]\displaystyle{ \R }[/math] that sends point [math]\displaystyle{ (a,b) }[/math] to [math]\displaystyle{ a+b }[/math]. The fiber of 5 under [math]\displaystyle{ f }[/math] are all the points on the straight line with equation [math]\displaystyle{ a+b=5 }[/math]. The fibers of [math]\displaystyle{ f }[/math] are that line and all the straight lines parallel to it, which form a partition of the plane [math]\displaystyle{ \R^2 }[/math].

More generally, if [math]\displaystyle{ f }[/math] is a linear map from some linear vector space [math]\displaystyle{ X }[/math] to some other linear space [math]\displaystyle{ Y }[/math], the fibers of [math]\displaystyle{ f }[/math] are affine subspaces of [math]\displaystyle{ X }[/math], which are all the translated copies of the null space of [math]\displaystyle{ f }[/math].

If [math]\displaystyle{ f }[/math] is a real-valued function of several real variables, the fibers of the function are the level sets of [math]\displaystyle{ f }[/math]. If [math]\displaystyle{ f }[/math] is also a continuous function and [math]\displaystyle{ y\in\R }[/math] is in the image of [math]\displaystyle{ f, }[/math] the level set [math]\displaystyle{ f^{-1}(y) }[/math] will typically be a curve in 2D, a surface in 3D, and, more generally, a hypersurface in the domain of [math]\displaystyle{ f. }[/math]

The fibers of [math]\displaystyle{ f }[/math] are the equivalence classes of the equivalence relation [math]\displaystyle{ \equiv_f }[/math] defined on the domain [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ x'\equiv_f x'' }[/math] if and only if [math]\displaystyle{ f(x') = f(x'') }[/math].

In topology

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

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

A function between topological spaces is called monotone if every fiber is a connected subspace of its domain. A function [math]\displaystyle{ f : \R \to \R }[/math] 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 [math]\displaystyle{ f }[/math] between topological spaces [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] whose fibers have certain special properties related to the topology of those spaces.

In algebraic geometry

In algebraic geometry, if [math]\displaystyle{ f : X \to Y }[/math] is a morphism of schemes, the fiber of a point [math]\displaystyle{ p }[/math] in [math]\displaystyle{ Y }[/math] is the fiber product of schemes [math]\displaystyle{ X \times_Y \operatorname{Spec} k(p) }[/math] where [math]\displaystyle{ k(p) }[/math] is the residue field at [math]\displaystyle{ p. }[/math]