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Fiber (mathematics)

## Summary

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

${\displaystyle f^{-1}(\{y\})=\{x\mathrel {:} f(x)=y\}}$

As an example of abuse of notation, this set is often denoted as ${\displaystyle f^{-1}(y)}$, which is technically incorrect since the inverse relation ${\displaystyle f^{-1}}$ of ${\displaystyle f}$ is not necessarily a function.

## Properties and applications

### In naive set theory

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

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

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

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

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

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

The fibers of ${\displaystyle f}$  are the equivalence classes of the equivalence relation ${\displaystyle \equiv _{f}}$  defined on the domain ${\displaystyle X}$  such that ${\displaystyle x'\equiv _{f}x''}$  if and only if ${\displaystyle f(x')=f(x'')}$ .

### In topology

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

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

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

### In algebraic geometry

In algebraic geometry, if ${\displaystyle f:X\to Y}$  is a morphism of schemes, the fiber of a point ${\displaystyle p}$  in ${\displaystyle Y}$  is the fiber product of schemes

${\displaystyle X\times _{Y}\operatorname {Spec} k(p)}$

where ${\displaystyle k(p)}$  is the residue field at ${\displaystyle p.}$