Fiber (mathematics)

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 ${\displaystyle \{y\}}$,^: 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 ${\displaystyle p.}$

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.