----- > [!proposition] Proposition. ([[decomposing Noetherian topological spaces]]) > Suppose $X$ is a [[Noetherian topological space|Noetherian]] [[topological space]]. Then every nonempty [[closed set|closed subset]] $Z \subset X$ can be expressed uniquely as a finite union $Z=Z_{1} \cup \dots \cup Z_{n}$ > of [[irreducible topological space|irreducible]] [[closed set|closed subsets]], none contained in any other. > [!proof]- Proof. ([[decomposing Noetherian topological spaces]]) > > The technique employed is called **Noetherian induction**. > > Let $\Sigma$ be the collection of nonempty closed subsets of $X$ that *cannot* be expressed as a finite union of irreducible closed subsets. WTS $\Sigma=\emptyset$. Suppose not: suppose $Y_{1} \in \Sigma$. If $Y_{1}$ properly contains another such, then choose one, and all it $Y_{2}$. If $Y_{2}$ properly contains another such, then choose one, and call it $Y_{3}$, and so on. The Noetherian hypothesis ([[ascending chain condition|descending chain condition]]) on $X$ ensures this must eventually stop: we must have some $Y_{r} \in \Sigma$ that is minimal by inclusion: $Y_{r}$ cannot be expressed as a finite union of irreducible closed subsets, but all of its proper nonempty closed subsets *can* be. $Y_{r}$ is not irreducible (by what we've just said); write $Y_{r}=Y' \cup Y''$ for $Y'$ and $Y''$ both proper closed subsets. Now we can write $Y'=Y'_{1} \cup \dots \cup Y'_{m'}$ and $Y''=Y_{1}'' \cup \dots \cup Y_{m''}''$, hence $Y_{r}=Y_{1}' \cup \dots \cup Y_{m}' \cup Y_{1}'' \cup \dots \cup Y_{m''}''$, contradicting that $Y_{r} \in \Sigma$. > > Thus each closed subset can be written as a finite union of irreducible closed subsets. We can assume that none of these contain any others, by discarding some of them. > > We now show uniqueness. > > - [ ] todo Vakil page 125; very easy but i am trying to move quickly right now ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```