Remark 5.28.5 (09Y2)—The Stacks project

Remark 5.28.5. Given a locally finite stratification $X = \coprod X_ i$ of a topological space $X$, we obtain a family of closed subsets $Z_ i = \bigcup _{j \leq i} X_ j$ of $X$ indexed by $I$ such that

\[ Z_ i \cap Z_ j = \bigcup \nolimits _{k \leq i, j} Z_ k \]

Conversely, given closed subsets $Z_ i \subset X$ indexed by a partially ordered set $I$ such that $X = \bigcup Z_ i$, such that every point has a neighbourhood meeting only finitely many $Z_ i$, and such that the displayed formula holds, then we obtain a locally finite stratification of $X$ by setting $X_ i = Z_ i \setminus \bigcup _{j < i} Z_ j$.

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Comment #2122 by UT on July 20, 2016 at 10:18

Why is $Z_i = \bigcup_{j \leq i} X_j$ closed? If the stratification is good, this holds by definition. But in general? Conversely, if we have a family of closed subsets Z_i satisfying the above equation then is it true that the induced stratification as described is good?

Comment #2141 by Johan on July 21, 2016 at 19:57

Note that $Z_i$ is the union of the closures of the $X_j$ for $j \leq i$ (this uses that we start with a stratifictation). Since the stratification is locally finite, then we see $Z_i$ is locally a finite union of closed sets. The answer to your second question is no. I suggest you make an example for yourself.

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2 comment(s) on Section 5.28: Partitions and stratifications

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