Fiber (mathematics)

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

# 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 $\$ under $f.$
# 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 $f^(y) = \,$ that is, the set of elements that get mapped to $y$ by the function. It is the preimage of the singleton $\.$ (One usually takes $y$ in the image of $f$ to avoid $f^(y)$ 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 $f^(f(x)).$ For example, the fibers of the projection map $\R^2\to\R$ 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 $y\in\R$ is in the image of $f,$ the level set $f^(y)$ 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
$X \times_Y \operatorname k(p)$
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 if every fiber is a connected subspace of its domain. A function $f : \R \to \R$ 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 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 .

See also

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

Citations

References

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Basic concepts in set theory
Mathematical relations