# Fiber (mathematics)

In mathematics, the term fiber (or fibre in 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 → 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 : XY be a map. The fiber of an element $y\in Y,$  commonly denoted by $f^{-1}(y),$  is defined as

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

That is, the fiber of y under f is the set of elements in the domain of f that are mapped to y.

The inverse image or preimage $f^{-1}(A)$  generalizes the concept of the fiber to subsets $A\subseteq Y$  of the codomain. The notation $f^{-1}(y)$  is still used to refer to the fiber, as the fiber of an element y is the preimage of the singleton set $\{y\}$ , as in $f^{-1}(\{y\})$ . That is, the fiber can be treated as a function from the codomain to the powerset of the domain: $f^{-1}:Y\to {\mathcal {P}}(X),$  while the preimage generalizes this to a function between powersets: $f^{-1}:{\mathcal {P}}(Y)\to {\mathcal {P}}(X).$

If f maps into the real numbers, so $y\in \mathbb {R}$  is simply a number, then the fiber $f^{-1}(y)$  is also called the level set of y under f: $L_{y}(f).$  If f is a continuous function and y is in the image of f, then the level set of y under f is a curve in 2D, a surface in 3D, and, more generally, a hypersurface of dimension d − 1.

### Fiber in algebraic geometry

In algebraic geometry, if f : X → 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.