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 $\{y\}$, 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.

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

$\{f^{-1}(y):y\in Y\}=\{\{x\in X:f(x)=y\}:y\in Y\}$

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 $x\in X$ is the set $f^{-1}(f(x)).$

For example, let $f$ be the function from ${\mathbb{ℝ}}^2$ to $\mathbb{ℝ}$ 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 ${\mathbb{ℝ}}^2$.

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 $y\in \mathbb{ℝ}$ 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 ${\equiv }_f$ defined on the domain $X$ such that $x^{\prime }{\equiv }_f{x}^{″}$ if and only if $f(x^{\prime })=f(x^{″})$.

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 $f:\mathbb{ℝ}\to \mathbb{ℝ}$ 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 $f$ 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 $f:X\to Y$ is a morphism of schemes, the fiber of a point $p$ in $Y$ is the fiber product of schemes $$X{\times }_Y\mathrm{Spec}k(p)$$ where $k(p)$ is the residue field at $p.$

See also

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