Can someone give a reasonably explicit example of an irreducible one-dimensional scheme with no closed points? 



One-dimensional scheme with no closed points
If $X$ is a 1-dimensional scheme with no closed points, let $x\in X$ be a point. Since $x$ is not closed, then there is some point $y\neq x$ in its closure. Since $y$ is not closed, there is a point $z\neq y$ in its closure. Now we have a chain $$\overline{\{z\}}\subset\overline{\{y\}}\subset\overline{\{x\}}$$ of irreducible closed subsets. This contradicts the assumption that $X$ is 1-dimensional.


More generally, the same argument shows any nonempty finite dimensional $T\_0$ space has a closed point (the $T\_0$ assumption is used to guarantee that the inclusions as above are strict).


