In mathematics, the term fiber (US English) or fibre (British English) can have two meanings, depending on the context:
Let be a function between sets.
The fiber of an element (or fiber over ) under the map is the set
The collection of all fibers for the function forms a partition of the domain The fiber containing an element is the set For example, the fibers of the projection map that sends to are the vertical lines, which form a partition of the plane.
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
In algebraic geometry, if is a morphism of schemes, the fiber of a point in is the fiber product of schemes
Every fiber of a local homeomorphism is a discrete subspace of its domain. If is a continuous function and if (or more generally, if ) is a T1 space then every fiber is a closed subset of
A function between topological spaces is called monotone 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 proper map 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 perfect map.