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In mathematics, the term **fiber** (US English) or **fibre** (British English) can have two meanings, depending on the context:

- In naive set theory, the
**fiber**of the element in the set under a map is the inverse image of the singleton under^{[1]} - In algebraic geometry, the notion of a fiber of a morphism of schemes must be defined more carefully because, in general, not every point is closed.

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 T_{1} 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*.

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