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Fiber (mathematics)

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(Redirected from Fiber (algebraic geometry)) Set of all points in a function's domain that all map to some single given point

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

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

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

Properties and applications

In naive set theory

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

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

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

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

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

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

In topology

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

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

A function between topological spaces is called monotone if every fiber is a connected subspace of its domain. A function f : R → R {\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 f {\displaystyle f} between topological spaces X {\displaystyle X} and Y {\displaystyle Y} whose fibers have certain special properties related to the topology of those spaces.

In algebraic geometry

In algebraic geometry, if f : X → Y {\displaystyle f:X\to Y} is a morphism of schemes, the fiber of a point p {\displaystyle p} in Y {\displaystyle Y} is the fiber product of schemes X × Y Spec ⁡ k ( p ) {\displaystyle X\times _{Y}\operatorname {Spec} k(p)} where k ( p ) {\displaystyle k(p)} is the residue field at p . {\displaystyle p.}

See also


References

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