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 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 $f : M \to N$ 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.^[1] For instance, the set of Liouville numbers is generic in the topological sense, but has Lebesgue measure zero.^[2]

^ Hunt, Brian R.; Kaloshin, Vadim Yu. (2010). Prevalence. Handbook of Dynamical Systems. Vol. 3. pp. 43–87. doi:10.1016/s1874-575x(10)00310-3. ISBN 9780444531414.
^ Oxtoby, John C. (1980). Measure and Category | SpringerLink. Graduate Texts in Mathematics. Vol. 2. doi:10.1007/978-1-4684-9339-9. ISBN 978-1-4684-9341-2.

[1] Hunt, Brian R.; Kaloshin, Vadim Yu. (2010). Prevalence. Handbook of Dynamical Systems. Vol. 3. pp. 43–87. doi:10.1016/s1874-575x(10)00310-3. ISBN 9780444531414.

[2] Oxtoby, John C. (1980). Measure and Category | SpringerLink. Graduate Texts in Mathematics. Vol. 2. doi:10.1007/978-1-4684-9339-9. ISBN 978-1-4684-9341-2.

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