I know that when a scheme $X$ is quasicompact, every point has a closed point in it's closure. This of course means that every nonempty quasicompact scheme has a closed point. If we drop the assumption that $X$ is quasicompact, it may no longer be true that $X$ contains a closed point. Is there a fairly easy example of this phenomenon?
I also want to better understand how to come up with these sorts of examples in the future, so if possible, could you briefly describe the general process you took to construct the example.
Best Answer
It turns out that Karl Schwede has a very nice example and explanation of how to construct a nonempty scheme with no closed points. Here is the link:
http://math.stanford.edu/~vakil/files/schwede03.pdf