Every non-generic point in a curve is closed? Let $X$ be a scheme integral scheme of dimension 1. If $X$ is affine, then it is clear that every non-generic point is closed. I wonder if this is true in general. If not, is it true if we suppose that $X$ is a curve? (A variety of dimension 1.)

If $x\in\overline{\{y\}}$, we say $x$ is a specialization of $y$ or $y$ is a generalization of $x$ and denote this $y\leadsto x$. Observe that in a scheme, a chain of specializations is in one-to-one correspondence with a chain of inclusions of irreducible closed subsets: in one direction, we take the closure of the points, while in the other, we take the generic points of the irreducible closed subsets. Therefore the dimension of a scheme is exactly the number of specializations in a maximal chain of nontrivial specializations.
Now suppose $X$ is irreducible of dimension one with generic point $\eta_X$. If $\overline{\{x\}}=\{x\}$, $x$ is closed. If $\overline{\{x\}}\neq\{x\}$, then let $y$ be some element distinct from $x$ in $\overline{\{x\}}$, and we get a chain of specializations $\eta_X\leadsto x\leadsto y$. As $x\neq y$, we must have $x=\eta_X$ otherwise we contradict the fact that $X$ has dimension one.
