In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the fiber of a point y in Y under a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 f : X → Y is the inverse image (also known as the preimage) of the singleton {y} under f, that is,

In a variant phrase, this is also called the fiber of f at y. It is also commonly denoted .

In various applications, this is also called:

1. The preimage of y under f, or the preimage of f at y. (Note that this terminology usually is used to refer to the preimages of subsets of Y; thus, to refer to the fiber of y one generally would call it the preimage of the singleton {y} under f)
2. The level set
   Level set
   In mathematics, a level set of a real-valued function f of n variables is a set of the formthat is, a set where the function takes on a given constant value c....
   
   of y under f, or the level set of f at y.

In algebraic geometry, the notion of a fiber of a morphism of schemes must be defined more carefully because in general, not every point is closed. In this case, if f : X → Y is a morphism of schemes, the fiber of a point p in Y is the fibered product where k(p) is the residue field at p.
In the same contexts, the spelling fibre is also seen.

See also

• Fibration
  Fibration
  In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...
  
  
• Fiber bundle
  Fiber bundle
  In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
  
  
• Fiber product
• Image (category theory)
  Image (category theory)
  Given a category C and a morphismf\colon X\to Y in C, the image of f is a monomorphism h\colon I\to Y satisfying the following universal property:#There exists a morphism g\colon X\to I such that f = hg....
  
  
• Image (mathematics)
  Image (mathematics)
  In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
  
  
• Inverse relation
  Inverse relation
  In mathematics, the inverse relation of a binary relation is the relation that occurs when you switch the order of the elements in the relation. For example, the inverse of the relation 'child of' is the relation 'parent of'...
  
  
• Kernel (mathematics)
  Kernel (mathematics)
  In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:* The kernel of a mapping is the set of elements that map to the zero element , as in kernel of a linear operator and kernel of a matrix...
  
  
• Preimage attack
  Preimage attack
  In cryptography, the preimage attack is a classification of attacks on hash functions for finding a message that has a specific hash value.There are two types of preimage attacks:...
  
  
• Relation
  Relation (mathematics)
  In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...