en>Mark viking: /* Statement of the theorem */ Added wl

2014-09-03T22:58:05Z

Statement of the theorem: Added wl

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	In [[mathematics]], the '''fiber''' of a point ''y'' in ''Y'' under a [[function (mathematics)\|function]] ''f'' : ''X'' → ''Y'' is the [[inverse image]] (also known as the preimage) of the [[singleton (mathematics)\|singleton]] {''y''} under ''f'', that is, <math>f^{-1}(\{y\})=\{x \in X : f(x) = y\}</math>		Friends call him Royal. What she enjoys performing is bottle tops collecting and she is trying to make it a occupation. I am a production and distribution officer. Alabama is exactly where he and his wife reside and he has everything that he needs there.<br><br>my web site - extended car warranty ([http://www.empredators.de/index.php?mod=users&action=view&id=10734 click through the up coming website])

	~~In a variant phrase, this is also called the '''fiber''' of ''f'' at ''y''~~. It is ~~also commonly denoted <math>f^{-1}(y)</math>.~~

	~~In various applications, this~~ is ~~also called:~~
	~~# The '''preimage''' of ''y'' under ''f'', or the '''preimage''' of ''f'' at ''y''. (Note that this terminology usually refers~~ 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'')~~
	~~# The '''[[level set]]''' of ''y'' under ''f'', or the '''level set''' of ''f'' at ''y''~~. ~~(Note that this terminology is only typically used if ''f'' maps into the real numbers~~ and ~~so ''y'' is simply a number~~. ~~If ''f''~~ is ~~a continuous function~~ and ~~if ''y'' is in the range of ''f'', then the '''[[level set]]''' of ''y'' under ''f'' is a curve in 2d or a surface in 3d,~~ and ~~generally a hypersurface of dimension ''d-1''.)~~

	~~In [[algebraic geometry]], the notion of a fiber of a [[morphism]] of [[Scheme (mathematics)\|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~~ <~~math~~>~~X\times_Y \mathrm{Spec}\, k(p)~~<~~/math~~> ~~where ''k''~~(~~''p'') is the residue field at ''p''~~.
	~~In the same contexts, the spelling '''fibre''' is also seen~~.

	== ~~See also~~ ==
	* [[Fibration]]
	* [[Fiber bundle]]
	* [[Fiber product]]
	* [[Image (category theory)]]
	* [[Image (mathematics)]]
	* [[Inverse relation]]
	* [[Kernel (algebra)\|Kernel (mathematics)]]
	* [[Preimage attack]]
	* [[Relation (mathematics)\|Relation]]

	~~[[Category:Set theory]]~~

	~~{{settheory-stub}}~~

en>SchreiberBike: Repairing links to disambiguation pages - You can help! - Multiplicity

2012-05-23T05:23:57Z

Repairing links to disambiguation pages - You can help! - Multiplicity

New page

In [[mathematics]], the '''fiber''' of a point ''y'' in ''Y'' under a [[function (mathematics)|function]] ''f'' : ''X'' → ''Y'' is the [[inverse image]] (also known as the preimage) of the [[singleton (mathematics)|singleton]] {''y''} under ''f'', that is, <math>f^{-1}(\{y\})=\{x \in X : f(x) = y\}</math>

In a variant phrase, this is also called the '''fiber''' of ''f'' at ''y''. It is also commonly denoted <math>f^{-1}(y)</math>.

In various applications, this is also called:
# The '''preimage''' of ''y'' under ''f'', or the '''preimage''' of ''f'' at ''y''. (Note that this terminology usually refers 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'')
# The '''[[level set]]''' of ''y'' under ''f'', or the '''level set''' of ''f'' at ''y''. (Note that this terminology is only typically used if ''f'' maps into the real numbers and so ''y'' is simply a number. If ''f'' is a continuous function and if ''y'' is in the range of ''f'', then the '''[[level set]]''' of ''y'' under ''f'' is a curve in 2d or a surface in 3d, and generally a hypersurface of dimension ''d-1''.)

In [[algebraic geometry]], the notion of a fiber of a [[morphism]] of [[Scheme (mathematics)|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 <math>X\times_Y \mathrm{Spec}\, k(p)</math> where ''k''(''p'') is the residue field at ''p''.
In the same contexts, the spelling '''fibre''' is also seen.

== See also ==
* [[Fibration]]
* [[Fiber bundle]]
* [[Fiber product]]
* [[Image (category theory)]]
* [[Image (mathematics)]]
* [[Inverse relation]]
* [[Kernel (algebra)|Kernel (mathematics)]]
* [[Preimage attack]]
* [[Relation (mathematics)|Relation]]

[[Category:Set theory]]

{{settheory-stub}}

Branching theorem - Revision history

en>Mark viking: /* Statement of the theorem */ Added wl

en>SchreiberBike: Repairing links to disambiguation pages - You can help! - Multiplicity