Fiber (mathematics)

Properties and applications

In elementary set theory

Any map ${\displaystyle f:X\to Y}$ defines a relation ${\displaystyle {\equiv }_f}$ on X, where ${\displaystyle x_1{\equiv }_f{x}_2}$ holds when ${\displaystyle f(x_1)=f(x_2)}$, and this is always an equivalence relation. The fibers of ${\displaystyle f}$ are the equivalence classes for this relation ${\displaystyle {\equiv }_f}$. Each such fiber is associated to the (by definition constant) value of f on the fiber, and if ${\displaystyle y\in Y}$ is that value (necessarily an element of the image set of f), then the fiber is the set ${\displaystyle f^{-1}(y)=\{x\in X:f(x)=y\}}$ of preimages of y under f, which is called the fiber of f over y. The set of all fibers of f forms is a partition of ${\displaystyle X}$. The mentioned correspondence defines a bijection from the quotient set ${\displaystyle X/{\equiv }_f}$ (the set of fibers of f) to the image set ${\displaystyle f(X)}$. Note that in case ${\displaystyle f}$ is not a surjective map, one only considers fibers over elements of the subset ${\displaystyle f(X)}$ of Y, thus avoiding the empty set, which cannot be an element of any partition, among the fibers of f. 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 ${\displaystyle f.}$

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 $${\displaystyle X{\times }_Y\mathrm{Spec}k(p)}$$ where ${\displaystyle k(p)}$ is the residue field at ${\displaystyle p.}$