This is a question I'm working on for algebraic geometry; I'm not sure, however, if it may be something that is true in a general topological space.
Let be a quasi-affine variety (an open subset of an affine variety). Suppose , and let
be a maximal chain of irreducible, closed subsets of .
Denote closures in by bars. Prove that
is a maximal chain of closed, irreducible subsets of .
So, I've got the closures to be closed (obviously..) and irreducible. I just need to show that the chain is maximal. I've been trying to prove the contrapositive, but I can't seem to get anywhere with it.