Fiber (mathematics)

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## Definitions

### Fiber in naive set theory

*y* under *f* is the set of elements in the domain of *f* that are mapped to *y*.
### Fiber in algebraic geometry

## See also

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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.

Let be a map. The **fiber** of an element commonly denoted by is defined as

That is, the fiber of

The inverse image or preimage generalizes the concept of the fiber to subsets of the codomain. The notation is still used to refer to the fiber, as the fiber of an element *y* is the preimage of the singleton set , as in . That is, the fiber can be treated as a function from the codomain to the powerset of the domain: while the preimage generalizes this to a function between powersets:

If *f* maps into the real numbers, so is simply a number, then the fiber is also called the **level set** of *y* under *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

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

where *k*(*p*) is the residue field at *p*.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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