In mathematics, the fiber (US English) or fibre (British English) of an element y under a function f is the preimage of the singleton set { y }, that is As an example of abuse of notation, this set is often denoted as f^{-1}(y), which is technically incorrect since the inverse relation f^{-1} of f is not necessarily a function.

Properties and applications

In naive set theory

If X and Y are the domain and image of f, respectively, then the fibers of f are the sets in which is a partition of the domain set X. Note that y must be restricted to the image set Y of f, since otherwise f^{-1}(y) would be the empty set which is not allowed in a partition. The fiber containing an element x\in X is the set For example, let f be the function from \R^2 to \R that sends point (a,b) to a+b. The fiber of 5 under f are all the points on the straight line with equation a+b=5. The fibers of f are that line and all the straight lines parallel to it, which form a partition of the plane \R^2. More generally, if f is a linear map from some linear vector space X to some other linear space Y, the fibers of f are affine subspaces of X, which are all the translated copies of the null space of f. If f is a real-valued function of several real variables, the fibers of the function are the level sets of f. If f is also a continuous function and y\in\R is in the image of f, the level set f^{-1}(y) will typically be a curve in 2D, a surface in 3D, and, more generally, a hypersurface in the domain of f. The fibers of f are the equivalence classes of the equivalence relation \equiv_f defined on the domain X such that if and only if.

In topology

In point set topology, one generally considers functions from topological spaces to topological spaces. If f is a continuous function and if Y (or more generally, the image set f(X)) is a T1 space then every fiber is a closed subset of X. In particular, if f is a local homeomorphism from X to Y, each fiber of f is a discrete subspace of X. A function between topological spaces is called if every fiber is a connected subspace of its domain. A function is monotone in this topological sense if and only if it is non-increasing or non-decreasing, which is the usual meaning of "monotone function" in real analysis. A function between topological spaces is (sometimes) called a if every fiber is a compact subspace of its domain. However, many authors use other non-equivalent competing definitions of "proper map" so it is advisable to always check how a particular author defines this term. A continuous closed surjective function whose fibers are all compact is called a. A fiber bundle is a function f between topological spaces X and Y whose fibers have certain special properties related to the topology of those spaces.

In algebraic geometry

In algebraic geometry, if f : X \to Y is a morphism of schemes, the fiber of a point p in Y is the fiber product of schemes where k(p) is the residue field at p.