Fiber in naive set theoryEdit

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

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

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

If ${\displaystyle f}$  is a real-valued function of several real variables, the fibers of the function are the level sets of ${\displaystyle f}$ . If ${\displaystyle f}$  is also a continuous function and ${\displaystyle y\in \mathbb {R} }$  is in the image of ${\displaystyle f,}$  the level set ${\displaystyle f^{-1}(y)}$  will typically be a curve in 2D, a surface in 3D, and, more generally, a hypersurface in the domain of ${\displaystyle f.}$ 

Fiber in algebraic geometryEdit

In algebraic geometry, if ${\displaystyle f:X\to Y}$  is a morphism of schemes, the fiber of a point ${\displaystyle p}$  in ${\displaystyle Y}$  is the fiber product of schemes

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

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