In mathematics, the fiber (US English) or fibre (British English) of an element
is the preimage of the singleton set
,[1]: p.69 that is As an example of abuse of notation, this set is often denoted as
, which is technically incorrect since the inverse relation
are the domain and image of
must be restricted to the image set
would be the empty set which is not allowed in a partition.
that sends point
are all the points on the straight line with equation
are that line and all the straight lines parallel to it, which form a partition of the plane
is a linear map from some linear vector space
, 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
the level set
will typically be a curve in 2D, a surface in 3D, and, more generally, a hypersurface in the domain of
In point set topology, one generally considers functions from topological spaces to topological spaces.
(or more generally, the image set
) is a T1 space then every fiber is a closed subset of
is a local homeomorphism from
A function between topological spaces is called monotone if every fiber is a connected subspace of its domain.
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
whose fibers have certain special properties related to the topology of those spaces.
In algebraic geometry, if
is a morphism of schemes, the fiber of a point
is the fiber product of schemes
is the residue field at