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.)
[1] For instance, the set of Liouville numbers is generic in the topological sense, but has Lebesgue measure zero.
This is the definition in the measure theory case specialized to a probability space.
In discrete mathematics, one uses the term almost all to mean cofinite (all but finitely many), cocountable (all but countably many), for sufficiently large numbers, or, sometimes, asymptotically almost surely.
Consequently, we rely on the stronger definition above which implies that the irrationals are typical and the rationals are not.
For applications, if a property holds on a residual set, it may not hold for every point, but perturbing it slightly will generally land one inside the residual set (by nowhere density of the components of the meagre set), and these are thus the most important case to address in theorems and algorithms.
A property of an irreducible algebraic variety X is said to be true generically if it holds except on a proper Zariski-closed subset of X, in other words, if it holds on a non-empty Zariski-open subset.
This definition agrees with the topological one above, because for irreducible algebraic varieties any non-empty open set is dense.
For example, by the Jacobian criterion for regularity, a generic point of a variety over a field of characteristic zero is smooth.
In characteristic zero, these equations are non-trivial, so they cannot be true for every point in the variety.
Consequently, the set of all non-regular points of X is a proper Zariski-closed subset of X.
Frequently this means that the ground field is uncountable and that the property is true except on a countable union of proper Zariski-closed subsets (i.e., the property holds on a dense Gδ set).
For instance, this notion of very generic occurs when considering rational connectedness.
Such results are frequently proved using the methods of limits of affine schemes developed in EGA IV 8.
A related concept in algebraic geometry is general position, whose precise meaning depends on the context.
For example, in the Euclidean plane, three points in general position are not collinear.
In computability and algorithmic randomness, an infinite string of natural numbers
[3] Some key properties are: 1-genericity is connected to the topological notion of "generic", as follows.
In particular, 1-generics are required to meet every dense open set (though this is a strictly weaker property, called weakly 1-generic).