Generic set

2020 Mathematics Subject Classification: Primary: 54E52 [MSN][ZBL]

Baire category

The terminology is often used (for instance in differential topology and in the theory of dynamical systems) for a set $E$ in a topological space $X$ which contains the intersection of a countable number of open dense subsets in $X$. The terminology is in general used when $X$ is a Baire space: in such spaces generic sets are dense. When some property $P$ which depends on a (topological) set of parameters $X$ holds for a generic subset, it is then commonly called a generic property or it is said that $P(x)$ holds for a generic $x$. A typical example is (a corollary of) Sard's theorem: Let $M$ and $N$ be two $C^r$ manifolds and $f:M\to N$ a surjective $C^r$ map. If $r> \max \{0, \dim M - \dim N\}$, then the set $\{f=x\}$ is a submanifold for a generic point $x$.

It is more common to call (a set containing) an intersection of countably many dense open sets residual or comeager, for its complement is meager, i.e. a union of countably many nowhere-dense sets (cf. Nowhere-dense set). Meager sets are considered topologically negligible (at least if $X$ is a Baire space), so that the Russian phrase "massive set" aptly describes residual sets. The Baire category theorem (cf. Baire theorem) says that in completely-metrizable spaces residual sets are dense; this is also true for locally compact spaces.

Logic

In mathematical logic and set theory the term "generic set" is also used. It refers to a "general" set of conditions which is consistent and is also such that every sentence from a specified set of sentences is decided.

The probably best-known example is found in forcing in set theory (cf. Forcing method). Here the full set of conditions is a partially ordered set and a generic set is then a filter which intersects all sets from a certain collection of dense sets. Another example occurs in model-theoretic forcing, where conditions are finite consistent sets of atomic sentences or negations of such.

In many instances the set of sentences to be decided is countable and the collection of maximally-consistent sets of conditions is a compact topological space. One can then invoke the Baire category theorem to conclude that the collection of generic sets is actually a dense generic set in the sense of the main article above.

Algebraic geometry

In algebraic (and analytic) geometry, when dealing with a scheme or pre-scheme with the Zariski topology a generic set usually means "containing a dense open set" or, what is often the same, the complement of a set of lower dimension. Thus, if one is dealing with a family of objects parametrized locally by a complex manifold or an analytic or algebraic variety, then the statement that a member of this family has a certain property generically or that a generic member of the family has the property means that the set of members of the family not having the property is contained in a subvariety of strictly lower dimension. For instance, the generic smoothness result says that if $f:X\to Y$ is a morphism of varieties over an algebraically closed field of characteristic zero and if $X$ is non-singular, then there is a non-empty open subset $U\subset Y$ such that $f: f^{-1} (U) \to U$ is smooth.

Cf. also Generic point.

References

[Ba]	R. Baire, "Leçons sur les fonctions discontinues, professées au collège de France" , Gauthier-Villars (1905) Zbl 36.0438.01
[Bar]	J. Barwise (ed.), Handbook of mathematical logic , North-Holland (1977) ((especially the article of D.A. Martin on Descriptive set theory)) MR0457132 Zbl 0623.03003 Zbl 0623.03002 Zbl 0528.03001 Zbl 0443.03001
[GH]	Ph.A. Griffiths, J. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) pp. 20 ff MR0507725 Zbl 0408.14001
[Ku]	K. Kunen, "Set theory" , North-Holland (1980) MR1567279 MR0597342 Zbl 0443.03021
[Ha]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 272 MR0463157 Zbl 0367.14001
[Ox]	J.C. Oxtoby, "Measure and category" , Springer (1971) MR0393403 0217.09201 Zbl 0217.09201