Definition 5.28.1. Let $X$ be a topological space. A partition of $X$ is a decomposition $X = \coprod X_ i$ into locally closed subsets $X_ i$. The $X_ i$ are called the parts of the partition. Given two partitions of $X$ we say one refines the other if the parts of one are unions of parts of the other.
5.28 Partitions and stratifications
Stratifications can be defined in many different ways. We welcome comments on the choice of definitions in this section.
Any topological space $X$ has a partition into connected components. If $X$ has finitely many irreducible components $Z_1, \ldots , Z_ r$, then there is a partition with parts $X_ I = \bigcap _{i \in I} Z_ i \setminus (\bigcup _{i \not\in I} Z_ i)$ whose indices are subsets $I \subset \{ 1, \ldots , r\} $ which refines the partition into connected components.
Definition 5.28.2. Let $X$ be a topological space. A good stratification of $X$ is a partition $X = \coprod X_ i$ such that for all $i, j \in I$ we have
\[ X_ i \cap \overline{X_ j} \not= \emptyset \Rightarrow X_ i \subset \overline{X_ j}. \]
Given a good stratification $X = \coprod _{i \in I} X_ i$ we obtain a partial ordering on $I$ by setting $i \leq j$ if and only if $X_ i \subset \overline{X_ j}$. Then we see that
\[ \overline{X_ j} = \bigcup \nolimits _{i \leq j} X_ i \]
However, what often happens in algebraic geometry is that one just has that the left hand side is a subset of the right hand side in the last displayed formula. This leads to the following definition.
Definition 5.28.3. Let $X$ be a topological space. A stratification of $X$ is given by a partition $X = \coprod _{i \in I} X_ i$ and a partial ordering on $I$ such that for each $j \in I$ we have The parts $X_ i$ are called the strata of the stratification.
\[ \overline{X_ j} \subset \bigcup \nolimits _{i \leq j} X_ i \]
We often impose additional conditions on the stratification. For example, stratifications are particularly nice if they are locally finite, which means that every point has a neighbourhood which meets only finitely many strata. More generally we introduce the following definition.
Definition 5.28.4. Let $X$ be a topological space. Let $I$ be a set and for $i \in I$ let $E_ i \subset X$ be a subset. We say the collection $\{ E_ i\} _{i \in I}$ is locally finite if for all $x \in X$ there exists an open neighbourhood $U$ of $x$ such that $\{ i \in I | E_ i \cap U \not= \emptyset \} $ is finite.
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 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 $X_ i = Z_ i \setminus \bigcup _{j < i} Z_ j$.
\[ Z_ i \cap Z_ j = \bigcup \nolimits _{k \leq i, j} Z_ k \]
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. $\square$
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. $\square$
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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