Generic point

In mathematics a generic point of a geometric object is a point with no special properties; a point which satsifies no conditions that do hold hold for every point in the space.

In topology
In general topology, a generic point of a topological space X is a point x such that the closure of the singleton set {x} is the whole of X: that is, x does not lie in any proper closed set in X.

In algebraic geometry
In algebraic geometry, a generic point of an algebraic variety is a point for which the coordinates have the property that the only polynomial relations that hold among them are the defining equations of the variety itself.

For example, consider the circle X2 + Y2 = 1 defined over the field of real numbers. The point


 * $$\left( \frac{2t}{1+t^2}, \frac{1-t^2}{1+t^2} \right) $$

with coefficients in the function field R(t) is generic, as it is on the circle but every polynomial relation between the coordinates is deducible from the relation X2 + Y2 = 1. On the other hand the point (3/5, 4/5) is not generic as it satisfies a rather trivial relation 5X=3 which does not hold in general.

The relation between the two uses of the term may be seen in the Zariski topology on an algebraic variety, where the closed sets are the zero sets of polynomial equations.