Fiber (mathematics) explained

under a function

is the preimage of the singleton set

\{y\}

, that is

f^-1(\{y\})=\{xl{:}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

is not necessarily a function.

Properties and applications

In naive set theory

and

are the domain and image of

, respectively, then the fibers of

are the sets in

\left\{f^-1(y)l{:}y\inY\right\} = \left\{\left\{x\inXl{:}f(x)=y\right\}l{:}y\inY\right\}

which is a partition of the domain set

. Note that

must be restricted to the image set

, since otherwise

f^-1(y)

would be the empty set which is not allowed in a partition. The fiber containing an element

x\inX

is the set

f^-1(f(x)).

For example, let

be the function from

\R²

that sends point

(a,b)

a+b

. The fiber of 5 under

are all the points on the straight line with equation

a+b=5

. The fibers of

are that line and all the straight lines parallel to it, which form a partition of the plane

\R²

More generally, if

is a linear map from some linear vector space

to some other linear space

, the fibers of

are affine subspaces of

, which are all the translated copies of the null space of

is a real-valued function of several real variables, the fibers of the function are the level sets of

. If

is also a continuous function and

y\in\R

is in the image of

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

The fibers of

are the equivalence classes of the equivalence relation

\equiv_f

defined on the domain

such that

x'\equiv_fx''

if and only if

f(x')=f(x'')

In topology

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

is a continuous function and if

(or more generally, the image set

f(X)

) is a T₁ space then every fiber is a closed subset of

In particular, if

is a local homeomorphism from

, each fiber of

is a discrete subspace of

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

between topological spaces

and

whose fibers have certain special properties related to the topology of those spaces.