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 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.)
- In measure theory, a generic property is one that holds almost everywhere, meaning "with probability 1", 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.
Other articles related to "generic property, generic":
... Sard's theorem If is a smooth function between smooth manifolds, then a generic point of N is not a critical value of f – critical values of f are a null set in N ... Jacobian criterion / generic smoothness A generic point of a variety over a field of characteristic zero is smooth ...
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