Generic point - Knowlpedia

Let $X$ be a scheme and $Z\subseteq X$ an irreducible closed subset. A point $\eta\in Z$ is a generic point of $Z$ if

\overline{\{\eta\}}=Z.

Every irreducible closed subset of a scheme has a unique generic point.

Example

If $A$ is an integral domain, the prime ideal $(0)$ is the generic point of $\operatorname{Spec}A$ , because its closure is the entire prime spectrum. Thus the point $(0)$ in $\operatorname{Spec}k[x]$ records properties holding away from every proper algebraic condition; it is not a missing numerical value of $x$ .

Remark

A generic point is an actual point of the Zariski topological space, not a probabilistically typical point or a metric limit point. It need not be closed.