Fiber (mathematics)

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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.

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