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In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic square matrix is invertible." As another example, a generic property of a space is a property that holds at "almost all" points of the space, as in the statement, "If f : M → N is a smooth function between smooth manifolds, then a generic point of N is not a critical value of f." (This is by Sard's theorem.)

Attributes	Values
rdf:type	Thing building
rdfs:label	Generische Eigenschaft (de) Generic property (en)
rdfs:comment	In der Mathematik werden Eigenschaften von Objekten als generisch bezeichnet, wenn sie in gewisser Weise typisch und nur in pathologischen Sonderfällen unzutreffend sind. Es gibt eine mathematisch klar definierte Verwendung des Begriffes „generisch“. Daneben wird die Bezeichnung aber auch informell verwandt um auszudrücken, dass eine Eigenschaft „meist“ oder „fast immer“ zutrifft. (de) In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic square matrix is invertible." As another example, a generic property of a space is a property that holds at "almost all" points of the space, as in the statement, "If f : M → N is a smooth function between smooth manifolds, then a generic point of N is not a critical value of f." (This is by Sard's theorem.) (en)
rdfs:seeAlso	Negligible set
dcterms:subject	Singularity theory Algebraic geometry
Wikipage page ID	12886758 (xsd:integer)
Wikipage revision ID	1090210433 (xsd:integer)
Link from a Wikipage to another Wikipage	Almost everywhere Almost surely Degeneration (algebraic geometry) John Wiley & Sons Nowhere dense set Computability theory Mathematics Measure theory General position Negligible set Function (mathematics) Configuration space (mathematics) Controllability Sard's theorem Line (geometry) Smooth function Sufficiently large Dense set Function space Meagre set Singularity theory Topology Gδ set Liouville number Square matrix Affine space Algebraic geometry Algebraic variety Algebraically independent Almost all Duality (mathematics) Base (topology) Null set Discrete mathematics Invertible matrix Baire space Baire space (set theory) Algebraic geometry Law of large numbers Zero of a function Polynomial Ground field Asymptotically almost surely Open set Rationally connected variety Real numbers Recursively enumerable set Scheme (mathematics) Smooth manifold Low (computability) Observability Zariski topology Linear time-invariant theory Whitney topologies Random graph Éléments de géométrie algébrique Springer-Verlag Residual set Residual subset Cocountable Cofinite Generic smoothness Algorithmic randomness dbr:Jacobian_criterion
sameAs	Generic property Generic property Generic property Generic property Generic property
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Expand_section dbt:Main_article dbt:Math dbt:Mvar dbt:Reflist dbt:See_also dbt:Short_description
has abstract	In der Mathematik werden Eigenschaften von Objekten als generisch bezeichnet, wenn sie in gewisser Weise typisch und nur in pathologischen Sonderfällen unzutreffend sind. Es gibt eine mathematisch klar definierte Verwendung des Begriffes „generisch“. Daneben wird die Bezeichnung aber auch informell verwandt um auszudrücken, dass eine Eigenschaft „meist“ oder „fast immer“ zutrifft. Häufig spricht man von generischen Eigenschaften von Funktionen oder Vektorfeldern, etwa in der oder in der Theorie der Gewöhnlichen Differentialgleichungen und dynamischen Systeme. In diesem Fall betrachtet man die Funktion als Element eines Funktionenraumes und meint, dass die entsprechende Eigenschaft generisch für Elemente dieses Funktionenraumes ist. (de) In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic square matrix is invertible." As another example, a generic property of a space is a property that holds at "almost all" points of the space, as in the statement, "If f : M → N is a smooth function between smooth manifolds, then a generic point of N is not a critical value of f." (This is by Sard's theorem.) There are many different notions of "generic" (what is meant by "almost all") in mathematics, with corresponding dual notions of "almost none" (negligible set); the two main classes are: * In measure theory, a generic property is one that holds almost everywhere, with the dual concept being null set, meaning "with probability 0". * In topology and algebraic geometry, a generic property is one that holds on a dense open set, or more generally on a residual set, with the dual concept being a nowhere dense set, or more generally a meagre set. There are several natural examples where those notions are not equal. For instance, the set of Liouville numbers is generic in the topological sense, but has Lebesgue measure zero. (en)
gold:hypernym	Properties
prov:wasDerivedFrom	wikipedia-en:Generic_property?oldid=1090210433&ns=0
page length (characters) of wiki page	11800 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Generic_property
is Link from a Wikipage to another Wikipage of	Antoine Song List of mathematical jargon Pathological (mathematics) René Thom Resultant Cubic graph

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