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. If $Z \in S$, then $Z$ is constructible in $X$ (Lemma 5.15.2). Moreover, $Z \cap Z' \in S$ for all $Z, Z' \in S$. Define a partial ordering on $S$ by inclusion. For $Z \in S$ set $X_ Z = Z \setminus \bigcup _{Z' < Z,\ Z' \in S} Z'$ to get a stratification $X = \coprod _{Z \in S} X_ Z$ satisfying the properties stated in the lemma. $\square$

Comment #6474 by Owen on

It appears that each $X_i$ is moreover locally constant constructible in the statement of the lemma. In the proof, $S$ seems to be doing notational double duty as a set of closed subsets of $X$ and as indices for such. Perhaps introduce a separate finite index set $\Lambda$ so that $S$ is the set of $Z_\lambda$ with $\lambda\in\Lambda$, so that in the end the stratification $X=\coprod_{\lambda\in\Lambda}X_\lambda$ will be indexed by $\Lambda$, with $X_\lambda=Z_\lambda\setminus\bigcup_{\lambda'<\lambda}Z_{\lambda'}$.

Comment #6475 by Owen on

*locally closed constructible

Comment #6550 by on

OK, I have fixed up the notation. See changes. The parts of a partition or the strata of a stratification are locally closed by Definitions 5.28.1 and 5.28.3.

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