**Generic Property**

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.)

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, 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.

Read more about Generic Property: Definitions: Measure Theory, Definitions: Topology, Genericity Results

### Other articles related to "generic property, generic":

**Generic Property**- Genericity Results

... 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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