Fiber (mathematics)

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 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.