# 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 ${\displaystyle \{y\}}$ 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 → Y be a map. The fiber of an element ${\displaystyle y\in Y}$, commonly denoted by ${\displaystyle f^{-1}(y)}$, is defined as

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

In various applications, this is also called:

The term level set is only used if f maps into the real numbers and so y is simply a number. If f is a continuous function and if 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 fibered product ${\displaystyle X\times _{Y}\mathrm {Spec} \,k(p)}$ where k(p) is the residue field at p.

## Terminological variance

The recommended practice is to use the terms fiber, inverse image, preimage, and level set as follows:

By abuse of language, the following terminology is sometimes used but should be avoided:

• the fiber of the map f at the element y
• the inverse image of the map f at the element y
• the preimage of the map f at the element y
• the level set of the point y under the map f.