The Stacks project

Lemma 5.28.8. Let $X$ be a Noetherian topological space. Any finite partition of $X$ can be refined by a finite good stratification.

Proof. Let $X = \coprod X_ i$ be a finite partition of $X$. Let $Z$ be an irreducible component of $X$. Since $X = \bigcup \overline{X_ i}$ with finite index set, there is an $i$ such that $Z \subset \overline{X_ i}$. Since $X_ i$ is locally closed this implies that $Z \cap X_ i$ contains an open of $Z$. Thus $Z \cap X_ i$ contains an open $U$ of $X$ (Lemma 5.9.2). Write $X_ i = U \amalg X_ i^1 \amalg X_ i^2$ with $X_ i^1 = (X_ i \setminus U) \cap \overline{U}$ and $X_ i^2 = (X_ i \setminus U) \cap \overline{U}^ c$. For $i' \not= i$ we set $X_{i'}^1 = X_{i'} \cap \overline{U}$ and $X_{i'}^2 = X_{i'} \cap \overline{U}^ c$. Then

\[ X \setminus U = \coprod X^ k_ l \]

is a partition such that $\overline{U} \setminus U = \bigcup X_ l^1$. Note that $X \setminus U$ is closed and strictly smaller than $X$. By Noetherian induction we can refine this partition by a finite good stratification $X \setminus U = \coprod _{\alpha \in A} T_\alpha $. Then $X = U \amalg \coprod _{\alpha \in A} T_\alpha $ is a finite good stratification of $X$ refining the partition we started with. $\square$

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