The Stacks project

Lemma 5.28.6. Let $X$ be a topological space. Let $X = \coprod X_ i$ be a finite partition of $X$. Then there exists a finite stratification of $X$ refining it.

Proof. Let $T_ i = \overline{X_ i}$ and $\Delta _ i = T_ i \setminus X_ i$. Let $S$ be the set of all intersections of $T_ i$ and $\Delta _ i$. (For example $T_1 \cap T_2 \cap \Delta _4$ is an element of $S$.) Then $S = \{ Z_ s\} $ is a finite collection of closed subsets of $X$ such that $Z_ s \cap Z_{s'} \in S$ for all $s, s' \in S$. Define a partial ordering on $S$ by inclusion. Then set $Y_ s = Z_ s \setminus \bigcup _{s' < s} Z_{s'}$ to get the desired stratification. $\square$

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