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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>
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| In a variant phrase, this is also called the '''fiber''' of ''f'' at ''y''. It is also commonly denoted <math>f^{-1}(y)</math>.
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| In various applications, this is also called:
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| # 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'')
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| # 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''.)
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| 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''.
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| In the same contexts, the spelling '''fibre''' is also seen.
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| == See also == | |
| * [[Fibration]]
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| * [[Fiber bundle]]
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| * [[Fiber product]]
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| * [[Image (category theory)]]
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| * [[Image (mathematics)]]
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| * [[Inverse relation]]
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| * [[Kernel (algebra)|Kernel (mathematics)]]
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| * [[Preimage attack]]
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| * [[Relation (mathematics)|Relation]]
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| [[Category:Set theory]]
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| {{settheory-stub}}
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