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

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

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.

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 $X_ i = Z_ i \setminus \bigcup _{j < i} Z_ j$.

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

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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