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

$\displaystyle Z_0\subsetneq Z_1\subsetneq \cdots \subsetneq Z_n$

be a maximal chain of irreducible, closed subsets of $\displaystyle Y$.

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

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

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

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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?