Branching theorem

In mathematics, the fiber of a point y in Y under a function f : X → Y is the inverse image (also known as the preimage) of the singleton {y} under f, that is, $f^{-1}(\{y\})=\{x\in X:f(x)=y\}$

In a variant phrase, this is also called the fiber of f at y. It is also commonly denoted $f^{-1}(y)$.

In various applications, this is also called:

1. The preimage of y under f, or the preimage of f at y. (Note that this terminology usually refers to the preimages of subsets of Y; thus, to refer to the fiber of y one generally would call it the preimage of the singleton {y} under f)
2. The level set of y under f, or the level set of f at y. (Note that this terminology is only typically 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 range of f, then the level set of y under f is a curve in 2d or a surface in 3d, and generally a hypersurface of dimension d-1.)

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. In this case, if f : X → Y is a morphism of schemes, the fiber of a point p in Y is the fibered product $X\times _{Y}\mathrm {Spec} \,k(p)$ where k(p) is the residue field at p. In the same contexts, the spelling fibre is also seen.

See also

• Fibration
• Fiber bundle
• Fiber product
• Image (category theory)
• Image (mathematics)
• Inverse relation
• Kernel (mathematics)
• Preimage attack
• Relation