Is a scheme with a single closed point affine?

Let $X$ be a quasi-compact, separated scheme with a single closed point.

Is $X$ necessarily affine, and thus isomorphic to the spectrum of a local ring?

I could not think of a counter-example; is there one?

Solution 1:

Well, I am answering my own question. As Cantlog points out, under the present assumptions, the question has a simple answer.

Let $X$ be a quasi-compact scheme with a single closed point $x_0$. Let $U$ be an open affine containing $x_0$. In a quasi-compact scheme $X$, every nonempty closed set contains a closed point of $X$. This implies that $X-U$ must be empty, since it does not contain $x_0$. So $X$ is affine. (Separatedness was not even necessary.)

I still do not know whether the question has a positive answer if $X$ is only assumed to be connected and separated. It seems like a more serious question.