Fiber (mathematics)

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 $$\{ y \}$$,  that is
 * $$f^{-1}(\{y\}) = \{ x \mathrel{:} f(x) = y \}$$

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

In naive 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 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 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.

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 $$p.$$