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In mathematics, the **fiber** (US English) or **fibre** (British English) of an element under a function is the preimage of the singleton set ,^{[1]}^{: p.69 } that is

As an example of abuse of notation, this set is often denoted as , which is technically incorrect since the inverse relation of is not necessarily a function.

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 .

In point set topology, one generally considers functions from topological spaces to topological spaces.

If is a continuous function and if (or more generally, the image set ) is a T_{1} space then every fiber is a closed subset of In particular, if is a local homeomorphism from to , each fiber of is a discrete subspace 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*.

A fiber bundle is a function between topological spaces and whose fibers have certain special properties related to the topology of those spaces.

In algebraic geometry, if is a morphism of schemes, the **fiber** of a point in is the fiber product of schemes

**^**Lee, John M. (2011).*Introduction to Topological Manifolds*(2nd ed.). Springer Verlag. ISBN 978-1-4419-7940-7.