Generic point

Remarks

A closed set with a generic point is irreducible: it is not the union of two smaller closed sets. In a scheme, every irreducible closed subset has a unique generic point, so the underlying space of a scheme is a sober space. Example: in $X = \operatorname{Spec} \Z$, the closure of $\set{(0)}$ is $V(0) = X$, because $(0)$ is contained in every prime ideal. So $(0)$ is the generic point of $X$. Non-example: the point $(p)$, for a prime $p$, is not a generic point of $X$, since $\set{(p)}$ is already closed. Non-example: $X$ itself has no generic point when $X = \operatorname{Spec} k[x, y]/(xy)$ for a field $k$, since $X$ is the union of the two lines $V(x)$ and $V(y)$.