Fiber (mathematics)

{{Short description|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 $y$ under a function $f$ is the preimage of the singleton set $\{ y \}$ ,{{rp|p.69}} that is

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

Properties and applications

=In elementary set theory=

If $X$ and $Y$ are the domain and image of $f$ , respectively, then the fibers of $f$ are the sets in

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

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

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

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

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 $y\in\R$ 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 $f.$

The fibers of $f$ are the equivalence classes of the equivalence relation $\equiv_f$ defined on the domain $X$ such that $x'\equiv_f x$ if and only if $f(x') = f(x$ ).

=In topology=

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

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

A function between topological spaces is called {{em|{{visible anchor|monotone}}}} if every fiber is a connected subspace of its domain. A function $f : \R \to \R$ 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 {{em|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 {{em|perfect map}}.

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

=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 $p.$

References

{{cite book|last=Lee|first=John M.|author-link=John M. Lee|publisher=Springer Verlag|year=2011|title=Introduction to Topological Manifolds|edition=2nd|isbn=978-1-4419-7940-7|url=https://www.springer.com/gp/book/9781441979391}}

Category:Basic concepts in set theory

Category:Mathematical relations