In naive set theory
If and are the domain and image of , respectively, then the fibers of are the sets in
which is a partition of the domain set . Note that must be restricted to the image set of , since otherwise would be the empty set which is not allowed in a partition. The fiber containing an element is the set
For example, let be the function from to that sends point to . The fiber of 5 under are all the points on the straight line with equation . The fibers of are that line and all the straight lines parallel to it, which form a partition of the plane .
More generally, if is a linear map from some linear vector space to some other linear space , the fibers of are affine subspaces of , which are all the translated copies of the null space of .
If is a real-valued function of several real variables, the fibers of the function are the level sets of . If is also a continuous function and is in the image of the level set will typically be a curve in 2D, a surface in 3D, and, more generally, a hypersurface in the domain of
The fibers of are the equivalence classes of the equivalence relation defined on the domain such that if and only if .