In mathematics, the term fiber ( US English) or fibre ( British English) can have two meanings, depending on the context:

1. In naive set theory, the fiber of the element ${\displaystyle y}$ in the set ${\displaystyle Y}$ under a map ${\displaystyle f:X\to Y}$ is the inverse image of the singleton ${\displaystyle \{y\}}$ under ${\displaystyle f.}$ ^[1]
2. In algebraic geometry, the notion of a fiber of a morphism of schemes must be defined more carefully because, in general, not every point is closed.

Definitions

Fiber in naive set theory

Let ${\displaystyle f:X\to Y}$ be a function between sets.

The fiber of an element ${\displaystyle y\in Y}$ (or fiber over ${\displaystyle y}$) under the map ${\displaystyle f}$ is the set

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

${\displaystyle y}$

${\displaystyle \{y\}.}$

${\displaystyle y}$

${\displaystyle f}$

${\displaystyle f^{-1}(y)}$

The collection of all fibers for the function ${\displaystyle f}$ forms a partition of the domain ${\displaystyle X.}$ The fiber containing an element ${\displaystyle x\in X}$ is the set ${\displaystyle f^{-1}(f(x)).}$ For example, the fibers of the projection map ${\displaystyle \mathbb {R} ^{2}\to \mathbb {R} }$ that sends ${\displaystyle (x,y)}$ to ${\displaystyle x}$ are the vertical lines, which form a partition of the plane.

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 ${\displaystyle f.}$

Fiber 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

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

${\displaystyle k(p)}$

${\displaystyle p.}$

Fibers in topology

Every fiber of a local homeomorphism is a discrete subspace of its domain. If ${\displaystyle f:X\to Y}$ is a continuous function and if ${\displaystyle Y}$ (or more generally, if ${\displaystyle f(X)}$) is a T₁ space then every fiber is a closed subset 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.

See also

Citations

References