Fiber (mathematics)

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 $y$ in the set $Y$ under a map $f:X\to Y$ is the inverse image of the singleton $\{y\}$ under $f.$
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 $f:X\to Y$ be a function between sets.

The fiber of an element $y\in Y$ (or fiber over $y$) under the map $f$ is the set $$f^{-1}(y)=\{x\in X:f(x)=y\},$$ that is, the set of elements that get mapped to $y$ by the function. It is the preimage of the singleton $\{y\}.$ (One usually takes $y$ in the image of $f$ to avoid $f^{-1}(y)$ being the empty set.)

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

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 $f.$

Fiber 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 $$X\times _{Y}\operatorname {Spec} k(p)$$ where $k(p)$ is the residue field at $p.$

Fibers in topology

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

See also

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

References

• Lee, John M. (2011). Introduction to Topological Manifolds (2nd ed.). Springer Verlag. ISBN 978-1-4419-7940-7.