Fiber (mathematics)
In mathematics, the term fiber (US English) or fibre (British English) can have two meanings, depending on the context: Definitions. Fiber in naive set theory. Let be a map. The fiber of an element formula_2 commonly denoted by formula_3 is defined as formula_4 That is, the fiber of under is the set of elements in the domain of that are mapped to . The inverse image or preimage formula_5 generalizes the concept of the fiber to subsets formula_6 of the codomain. The notation formula_7 is still used to refer to the fiber, as the fiber of an element is  the preimage of the singleton set formula_1, as in formula_9. That is, the fiber can be treated as a function from the codomain to the powerset of the domain: formula_10 while the preimage generalizes this to a function between powersets: formula_11 If maps into the real numbers, so formula_12 is simply a number, then the fiber formula_7 is also called the "level set" of under : formula_14 If is a continuous function and is in the image of , then the level set of under is a curve in 2D, a surface in 3D, and, more generally, a hypersurface of dimension Fiber  in algebraic geometry. In algebraic geometry, if is a morphism of schemes, the fiber of a point in is the fiber product of schemes where formula_16 is the residue field at .