generic property

generic property (plural generic properties)

1. Used other than figuratively or idiomatically: see generic,‎ property.
   • 1984, D. W. Hamlyn, Metaphysics, Cambridge University Press, page 82:

     To argue in that way, however, is not to compare like with like; it is to contrast the generic property of extension with the more specific properties of hardness, smell and colour. It is not after all clear that it loses the generic properties of smell, colour and perhaps solidity, as opposed to the more specific forms of those properties that the unmelted wax may have.

2. (mathematics, measure theory) A property that is true almost everywhere in a given set (i.e., the set of points at which the property is not true is either of measure zero or a subset of a set of measure zero).
3. (topology, algebraic geometry) A property that is true in some dense open subset of a given set.
   • 1989, Thomas S. Parker, Leon O. Chua, Appendix C: Differential Topology, Practical Numerical Algorithms for Chaotic Systems, Springer, Softcover reprint, page 314,

     A property P that refers to members of a set Y is a generic property if the subset of Y whose members exhibit property P contains a dense open subset of Y.

     Remark: If P₁ and P₂ are generic properties of a set Y, then so is the property "P₁ and P₂."

   • 2001, Constantin I. Chueshov (translator), Igor Chueshov, Introduction to the Theory of Infinite-dimensional Dissipative Systems, [1999, Russian edition] Acta Scientific Publishing House, page 117,

     It should be noted that if a property of a dynamical system holds for the parameters from an open and dense set in the corresponding space, then it is frequently said that this property is a generic property.

     However, it should be kept in mind that the generic property is not the one that holds almost always. […] Therefore, it should be remembered that generic properties are quite frequently encountered and stay stable during small perturbations of the properties of a system.

   • 2010, H. W. Broer, F. Takens, “Chapter 1: Preliminaries of Dynamical Systems Theory”, in B. Hasselblatt, H. W. Broer, F. Takens, editors, Handbook of Dynamical Systems, volume 3, Elsevier (North-Holland), page 24:

     We now formulate the first generic property concerning the singularities of a Hamiltonian function.

The two definitions are close but not identical. By way of example, the set of Liouville numbers is a dense open subset of the real numbers that has Lebesgue measure zero. Thus, "being a Liouville number" is a generic property in the topological sense but not in the measure theory sense.

• generic point