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Fiber (mathematics) - Wikipedia

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

{\displaystyle f^{-1}(\{y\})=\{x\mathrel {:} f(x)=y\}}

As an example of abuse of notation, this set is often denoted as {\displaystyle f^{-1}(y)}, which is technically incorrect since the inverse relation {\displaystyle f^{-1}} of {\displaystyle f} is not necessarily a function.

Properties and applications

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In naive set theory

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If {\displaystyle X} and {\displaystyle Y} are the domain and image of {\displaystyle f}, respectively, then the fibers of {\displaystyle f} are the sets in

{\displaystyle \left\{f^{-1}(y)\mathrel {:} y\in Y\right\}\quad =\quad \left\{\left\{x\in X\mathrel {:} f(x)=y\right\}\mathrel {:} y\in Y\right\}}

which is a partition of the domain set {\displaystyle X}. Note that {\displaystyle y} must be restricted to the image set {\displaystyle Y} of {\displaystyle f}, since otherwise {\displaystyle f^{-1}(y)} would be the empty set which is not allowed in a partition. The fiber containing an element {\displaystyle x\in X} is the set {\displaystyle f^{-1}(f(x)).}

For example, let {\displaystyle f} be the function from {\displaystyle \mathbb {R} ^{2}} to {\displaystyle \mathbb {R} } that sends point {\displaystyle (a,b)} to {\displaystyle a+b}. The fiber of 5 under {\displaystyle f} are all the points on the straight line with equation {\displaystyle a+b=5}. The fibers of {\displaystyle f} are that line and all the straight lines parallel to it, which form a partition of the plane {\displaystyle \mathbb {R} ^{2}}.

More generally, if {\displaystyle f} is a linear map from some linear vector space {\displaystyle X} to some other linear space {\displaystyle Y}, the fibers of {\displaystyle f} are affine subspaces of {\displaystyle X}, which are all the translated copies of the null space of {\displaystyle f}.

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.}

The fibers of {\displaystyle f} are the equivalence classes of the equivalence relation {\displaystyle \equiv _{f}} defined on the domain {\displaystyle X} such that {\displaystyle x'\equiv _{f}x''} if and only if {\displaystyle f(x')=f(x'')}.

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

If {\displaystyle f} is a continuous function and if {\displaystyle Y} (or more generally, the image set {\displaystyle f(X)}) is a T1 space then every fiber is a closed subset of {\displaystyle X.} In particular, if {\displaystyle f} is a local homeomorphism from {\displaystyle X} to {\displaystyle Y}, each fiber of {\displaystyle f} is a discrete subspace 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.

A fiber bundle is a function {\displaystyle f} between topological spaces {\displaystyle X} and {\displaystyle Y} whose fibers have certain special properties related to the topology of those spaces.

In algebraic geometry

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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 {\displaystyle X\times _{Y}\operatorname {Spec} k(p)} where {\displaystyle k(p)} is the residue field at {\displaystyle p.}