TITLE: Example of an affine scheme where closed points aren't dense.
QUESTION [5 upvotes]: I'm looking for an example of an affine scheme where closed points aren't dense. It's easy to show (using Hilbert's Nullstellensatz) that if $A$ is a finitely generated algebra over a field, then the closed points of $\operatorname {Spec} A$ are dense. Therefore, I suppose that I can find an example where $A$ is not finitely generated. Specifically, I'm looking for an open set in $k[x_1,x_2,...]$ that doesn't contain any closed point. But then, how can it contain a prime ideal at all?

REPLY [2 votes]: In general, if $A$ is not Jacobson (that is, if the Jacobson radical of $A$ is not equal to the nilradical) then the closed points of $\mbox{Spec}(A)$ are not dense. The converse is also true!