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## Summary

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

1. In naive set theory, the fiber of the element $y$ in the set $Y$ under a map $f:X\to Y$ is the inverse image of the singleton $\{y\}$ under $f.$ 2. 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.

## Definitions

### Fiber in naive set theory

Let $f:X\to Y$  be a function between sets.

The fiber of an element $y\in Y$  (or fiber over $y$ ) under the map $f$  is the set

$f^{-1}(y)=\{x\in X:f(x)=y\},$

that is, the set of elements that get mapped to $y$  by the function. It is the preimage of the singleton $\{y\}.$  (One usually takes $y$  in the image of $f$  to avoid $f^{-1}(y)$  being the empty set.)

The collection of all fibers for the function $f$  forms a partition of the domain $X.$  The fiber containing an element $x\in X$  is the set $f^{-1}(f(x)).$  For example, the fibers of the projection map $\mathbb {R} ^{2}\to \mathbb {R}$  that sends $(x,y)$  to $x$  are the vertical lines, which form a partition of the plane.

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 $y\in \mathbb {R}$  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 $f.$

### Fiber 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

$X\times _{Y}\operatorname {Spec} k(p)$

where $k(p)$  is the residue field at $p.$

## Fibers in topology

Every fiber of a local homeomorphism is a discrete subspace of its domain. If $f:X\to Y$  is a continuous function and if $Y$  (or more generally, if $f(X)$ ) is a T1 space then every fiber is a closed subset of $X.$

A function between topological spaces is called monotone if every fiber is a connected subspace of its domain. A function $f:\mathbb {R} \to \mathbb {R}$  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.