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 Y be a quasi-affine variety (an open subset of an affine variety). Suppose \mathrm{dim}(Y)=n<\infty, and let

Z_0\subsetneq Z_1\subsetneq \cdots \subsetneq Z_n

be a maximal chain of irreducible, closed subsets of Y.

Denote closures in \overline{Y} by bars. Prove that

\overline{Z_0}\subsetneq \overline{Z_1}\subsetneq \cdots \subsetneq \overline{Z_n}

is a maximal chain of closed, irreducible subsets of \overline{Y}.

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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.

Any ideas?