Closed point

Definition

For $\operatorname{Spec}F$ with $F$ a field, the unique point is closed. In $\operatorname{Spec}k[x]$, the points $(x-a)$ for $a\in k$ are the familiar geometric points when $k$ is algebraically closed, while $(0)$ is the generic point and is not closed.

A point $x$ of a scheme $X$ is a closed point if its singleton is closed:

For an affine scheme $X=\operatorname{Spec}A$, the point corresponding to a prime ideal $\mathfrak p$ is closed if and only if $\mathfrak p$ is maximal. Thus the closed points of $\operatorname{Spec}A$ are precisely the maximal ideals of $A$.

If $X$ is of finite type over an algebraically closed field $k$, its closed points have residue field $k$ and correspond to the classical geometric points described by coordinates.