The Stacks project

Lemma 5.28.7. Let $X$ be a topological space. Suppose $X = T_1 \cup \ldots \cup T_ n$ is written as a union of constructible subsets. There exists a finite stratification $X = \coprod X_ i$ with each $X_ i$ constructible such that each $T_ k$ is a union of strata.

Proof. By definition of constructible subsets, we can write each $T_ i$ as a finite union of $U \cap V^ c$ with $U, V \subset X$ retrocompact open. Hence we may assume that $T_ i = U_ i \cap V_ i^ c$ with $U_ i, V_ i \subset X$ retrocompact open. Let $S$ be the finite set of closed subsets of $X$ consisting of $\emptyset , X, U_ i^ c, V_ i^ c$ and finite intersections of these. Write $S = \{ Z_ s\} $. If $s \in S$, then $Z_ s$ is constructible (Lemma 5.15.2). Moreover, $Z_ s \cap Z_{s'} \in S$ 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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