I have to annoy you just one further time with these closed subset stories.
I am trying to make rigorous a proof, in which the author tries to show an equality of closed subsets $Y$ and $Z$ of an abelian variety $A$.
He shows that whenever you take a closed (!) point on $A$ which is contained in $Z$, then it is also contained in $Y$. Then he concludes that $Z$ is a subset of $Y$. Why can he do this?
I tried to make it rigorous by thinking locally, but it didn't work because the jacobson radical of an ideal is not the radical of an ideal.
I tried it with general topology, but I didn't see what the essential point should be.
Surely one has to remark that the closed points on $A$ are dense in $A$.
Furthermore, I can remark that the set $Y$ is actually the graph of the inversion morphism $i:A\rightarrow A$. The set $Z$ is the support of a line bundle.
Thanks a lot!
For any subset $S\subset A \;$ of a scheme $A$ , call $S^{cl}$ the set of closed points of $A$ contained in $S$.
Then :
Theorem
Let $k$ be a field. If the $k$- scheme $A$ is locally algebraic, then $A^{cl}$ is very dense in $A$.
Explanations
i) That $A$ is locally algebraic means that $A$ can be covered by open sets isomorphic to $Spec(R)$, with $R$ a finitely generated $k$-algebra .
ii) That $A^{cl}$ is very dense in $A$ means that for all closed subsets $Z\subset A$, the subset $Z^{cl}$ is dense in $Z$, that is $\bar {Z^{cl}}=Z$
Application to your problem The result you want is now immediate: since your hypothesis is $Z^{cl} \subset Y $, you immediately deduce $\bar{ Z^{cl}}=Z\ \subset \bar Y=Y$
As you see the fact that the base field is algebraically closed or of characteristic zero is irrelevant, and similarly the hypothesis that $A$ is abelian or even that $A$ is complete is also supefluous!