Fiber (mathematics)

In mathematics, the term fiber (or fibre in British English) can have two meanings, depending on the context:

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

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.

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.

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