---- > [!definition] Definition. ([[Noetherian topological space]]) > A [[topological space]] $X$ is said to be **Noetherian** if it satisfies the [[ascending chain condition|descending chain condition]] for [[closed set|closed subsets]]. ^definition > [!basicexample] > If $A$ is a [[Noetherian ring|Noetherian]] [[ring]] — in particular [[ascending chain - maximality characterization of Noetherian modules|satisfies the a.c.c.]] — then [[Zariski topology on a ring spectrum|the space]] $X=\text{Spec }A$ is Noetherian. Indeed, since $V(\cdot)$ reverses inclusions, any descending chain of closed subsets $V(I_{1}) \supset V(I_{2}) \supset \dots \supset V(I_{n}) \supset\dots$ > arises from an *ascending* chain of [[ideal|ideals]] $I_{1} \subset I_{2} \subset \dots \subset I_{n} \subset \dots$ > The latter eventually stabilizes, so the former does too. > [!basicproperties] > - [[decomposing Noetherian topological spaces]] > - If $X_{1}$ and $X_{2}$ are Noetherian topological spaces, so is $X_{1} \cup X_{2}$. > > [!proof]- > > Indeed, consider a descending chain of closed subsets in $X_{1} \cup X_{2}$ $Z_{1} \supset Z_{2} \supset\dots \supset Z_{n} \supset \dots$ > > Then the descending chains $X_{1} \cap Z_{1} \supset X_{1} \cap Z_{2} \supset \dots \supset X_{1} \cap Z_{n} \supset \dots$ > > and $X_{2} \cap Z_{1} \supset X_{2} \cap Z_{2} \supset \dots \supset X_{2} \cap Z_{n} \supset \dots$ > > stabilize as chains in $X_{1}$ and $X_{2}$ respectively, say, at indices $r_{1}$, $r_{2}$. Let $r:=\max(r_{1},r_{2})$. Then since $Z_{n}=(Z_{n} \cap X_{1}) \cup (Z_{n} \cap X_{2})$, the chain $Z_{1} \supset Z_{2} \supset\dots \supset Z_{n} \supset \dots$ > > stabilizes past $r$. > > > ^7a38e7 ---- #### ---- #### 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 > ```