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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In mathematics, a function between topological spaces is called proper if s of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism. There are several competing definitions of a "proper function". Some authors call a function between two topological spaces if the of every compact set in is compact in Other authors call a map if it is continuous and ; that is if it is a continuous closed map and the preimage of every point in is compact.
In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real values, while the "output", also called the "value of the function", may be real or complex.
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function is open if for any open set in the is open in Likewise, a closed map is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa. Open and closed maps are not necessarily continuous.
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