# Fiber (mathematics)

In mathematics, the term **fiber** (or **fibre** in 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*→*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* → *Y* be a map. The **fiber** of an element , commonly denoted by , is defined as

In various applications, this is also called:

- the inverse image of under the map
*f* - the preimage of under the map
*f* - the level set of the function
*f*at the point*y*.

- the inverse image of under the map

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 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:

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

- the

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*.

- the

## See also

- Fibration
- Fiber bundle
- Fiber product
- Image (category theory)
- Image (mathematics)
- Inverse relation
- Kernel (mathematics)
- Level set
- Preimage
- Relation
- Zero set