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Let $X$ be a noetherian integral scheme. Let $Z\subset X$ be a set of closed points in $X$.

What is the dimension of the closure of $Z$?

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In general $Z$ is not open in $\overline{Z}$ (think about schemes of finite type over a field). – user18119 Dec 10 '11 at 22:10
Ow I was confuse there. You're right. I could consider the set of closed points $Z$ in a curve. Its closure is the whole set. Moreover, $Z$ is not open in the curve, because the generic point is not closed. – Lucci Dec 11 '11 at 0:12
You mean the dimension of the closure compared to $\dim Z$ ? – user18119 Dec 12 '11 at 22:33

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