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	~~I would like to introduce myself to you~~, ~~I am Jayson Simcox but I~~ [~~http://modenpeople~~.co.kr/~~modn~~/~~qna~~/~~292291 phone psychic readings~~] ~~don't like when people~~ [~~http~~://~~alles-herunterladen~~.de/~~excellent-advice-~~for~~-picking-~~the-ideal~~-hobby/ love psychic~~] ~~use my full~~ name. ~~Doing ballet~~ is ~~something she would by no means give up~~. ~~Distributing manufacturing~~ is exactly ~~where my main earnings comes from~~ and ~~it's some thing I really enjoy. For years she's been residing in Kentucky~~ but ~~her husband desires them to transfer~~.<br><br>~~My web page; psychic love readings~~ ([~~http~~://~~www~~.~~edmposts~~.~~com~~/~~build-a-beautiful-organic-garden-using-these-ideas~~/ ~~www~~.~~edmposts~~.~~com~~])		In [[mathematics]], a '''Noetherian topological space''' is a [[topological space]] in which closed subsets satisfy the [[descending chain condition]]. Equivalently, we could say that the open subsets satisfy the [[ascending chain condition]], since they are the complements of the closed subsets. It can also be shown to be equivalent that every open subset of such a space is [[compact space\|compact]], and in fact the seemingly stronger statement that ''every'' subset is compact.

			== Definition ==
			A [[topological space]] <math>X</math> is called '''Noetherian''' if it satisfies the [[descending chain condition]] for [[closed subset]]s: for any [[sequence]]

			:<math> Y_1 \supseteq Y_2 \supseteq \cdots </math>

			of closed subsets <math>Y_i</math> of <math>X</math>, there is an [[integer]] <math>m</math> such that <math>Y_m=Y_{m+1}=\cdots.</math>

			== Relation to compactness ==
			The Noetherian condition can be seen as a strong [[compact space\|compactness]] condition:
			*Every Noetherian topological space is compact.
			*A topological space <math>X</math> is Noetherian if and only if every [[Subspace topology\|subspace]] of <math>X</math> is compact. (i.e. <math>X</math> is hereditarily compact).

			== Noetherian topological spaces from algebraic geometry ==

			Many examples of Noetherian topological spaces come from [[algebraic geometry]], where for the [[Zariski topology]] an [[irreducible set]] has the intuitive property that any closed proper subset has smaller dimension. Since dimension can only 'jump down' a finite number of times, and [[algebraic set]]s are made up of finite unions of irreducible sets, descending chains of Zariski closed sets must eventually be constant.

			A more algebraic way to see this is that the associated [[ideal (ring theory)\|ideals]] defining algebraic sets must satisfy the [[ascending chain condition]]. That follows because the rings of algebraic geometry, in the classical sense, are [[Noetherian ring]]s. This class of examples therefore also explains the name.

			If ''R'' is a commutative Noetherian ring, then Spec(''R''), the [[prime spectrum]] of ''R'', is a Noetherian topological space. More generally, a [[Noetherian scheme]] is a Noetherian topological space. The converse does not hold, since Spec(''R'') of an one dimensional valuation domain ''R'' consists of exactly two points and therefore is Noetherian, but there are examples of such rings which are not Noetherian.

			== Example ==
			The space <math>\mathbb{A}^n_k</math> (affine <math>n</math>-space over a [[Field (mathematics)\|field]] <math>k</math>) under the [[Zariski topology]] is an example of a Noetherian topological space. By properties of the [[right ideal\|ideal]] of a subset of <math>\mathbb{A}^n_k</math>, we know that if

			:<math>Y_1 \supseteq Y_2 \supseteq Y_3 \supseteq \cdots</math>

			is a descending chain of Zariski-closed subsets, then

			:<math>I(Y_1) \subseteq I(Y_2) \subseteq I(Y_3) \subseteq \cdots</math>

			is an ascending chain of ideals of <math>k[x_1,\ldots,x_n].</math> Since <math>k[x_1,\ldots,x_n]</math> is a Noetherian ring, there exists an integer <math>m</math> such that

			:<math>I(Y_m)=I(Y_{m+1})=I(Y_{m + 2})=\cdots.</math>

			Since <math>V(I(Y))</math> is the closure of ''Y'' for all ''Y'', <math>V(I(Y_i))=Y_i</math> for all <math>i.</math> Hence

			:<math>Y_m=Y_{m+1}=Y_{m + 2}=\cdots</math> as required.

			==References==
			*{{Hartshorne AG}}

			{{PlanetMath attribution\|id=3465\|title=Noetherian topological space}}

			[[Category:Algebraic geometry]]
			[[Category:Properties of topological spaces]]
			[[Category:Scheme theory]]
			[[Category:Wellfoundedness]]

en>SmackBot: remove Erik9bot category,outdated, tag and general fixes, added orphan tag

2009-12-16T11:58:21Z

remove Erik9bot category,outdated, tag and general fixes, added orphan tag

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I would like to introduce myself to you, I am Jayson Simcox but I [http://modenpeople.co.kr/modn/qna/292291 phone psychic readings] don't like when people [http://alles-herunterladen.de/excellent-advice-for-picking-the-ideal-hobby/ love psychic] use my full name. Doing ballet is something she would by no means give up. Distributing manufacturing is exactly where my main earnings comes from and it's some thing I really enjoy. For years she's been residing in Kentucky but her husband desires them to transfer.<br><br>My web page; psychic love readings ([http://www.edmposts.com/build-a-beautiful-organic-garden-using-these-ideas/ www.edmposts.com])

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en>Addbot: Bot: Removing Orphan Tag (Nolonger an Orphan) (Report Errors)

en>SmackBot: remove Erik9bot category,outdated, tag and general fixes, added orphan tag