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