Fiber in naive set theory
Let f : X → Y be a map. The fiber of an element commonly denoted by is defined as
That is, the fiber of y under f is the set of elements in the domain of f that are mapped to y.
The inverse image or preimage generalizes the concept of the fiber to subsets of the codomain. The notation is still used to refer to the fiber, as the fiber of an element y is the preimage of the singleton set , as in . That is, the fiber can be treated as a function from the codomain to the powerset of the domain: while the preimage generalizes this to a function between powersets:
If f maps into the real numbers, so is simply a number, then the fiber is also called the level set of y under f: If f is a continuous function and 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.