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!