Noetherian space: Difference between revisions
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In [[topology]], a '''Noetherian space''' is a [[topological space]] satisfying the [[descending chain condition]] on [[closed set]]s. | In [[topology]], a '''Noetherian space''' is a [[topological space]] satisfying the [[descending chain condition]] on [[closed set]]s. | ||
A closed set in a Noetherian space is again Noetherian with respect to the [[induced topology]]. | A closed set in a Noetherian space is again Noetherian with respect to the [[induced topology]]. | ||
The motivating example, and origin of the terminology, is that of the [[Zariski topology]] on an [[affine scheme]], where the closed sets are precisely the [[zero set]]s of [[ideal]]s of the corresponding ring ''A'' (in an order-reversing correspondence). The space is Noetherian if and only if ''A'' is a [[Noetherian ring]]. | The motivating example, and origin of the terminology, is that of the [[Zariski topology]] on an [[affine scheme]], where the closed sets are precisely the [[zero set]]s of [[ideal]]s of the corresponding ring ''A'' (in an order-reversing correspondence). The space is Noetherian if and only if ''A'' is a [[Noetherian ring]].[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 12:00, 26 September 2024
In topology, a Noetherian space is a topological space satisfying the descending chain condition on closed sets.
A closed set in a Noetherian space is again Noetherian with respect to the induced topology.
The motivating example, and origin of the terminology, is that of the Zariski topology on an affine scheme, where the closed sets are precisely the zero sets of ideals of the corresponding ring A (in an order-reversing correspondence). The space is Noetherian if and only if A is a Noetherian ring.