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Concept
Fiber (mathematics)
Summary
In mathematics, the term fiber (US English) or fibre (British English) can have two meanings, depending on the context:
In naive set theory, the fiber of the element in the set under a map is the of the singleton under
In algebraic geometry, the notion of a fiber of a morphism of schemes must be defined more carefully because, in general, not every is closed.
Let be a function between sets.
The fiber of an element (or fiber over ) under the map is the set that is, the set of elements that get mapped to by the function. It is the of the singleton (One usually takes in the of to avoid being the empty set.)
The collection of all fibers for the function forms a partition of the domain The fiber containing an element is the set For example, the fibers of the projection map that sends to are the vertical lines, which form a partition of the plane.
If 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 is in the 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 algebraic geometry, if is a morphism of schemes, the fiber of a in is the fiber product of schemes
where is the residue field at
Every fiber of a local homeomorphism is a discrete subspace of its domain.
If is a continuous function and if (or more generally, if ) is a T1 space then every fiber is a closed subset of
A function between topological spaces is called if every fiber is a connected subspace of its domain. A function 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 .
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