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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).2011-12-10
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    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.2011-12-11
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    You mean the dimension of the closure compared to $\dim Z$ ?2011-12-12

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