Noetherian space

E29919

mathematical concept topological notion

A Noetherian space is a topological space in which every descending chain of closed subsets stabilizes, mirroring the finiteness conditions of Noetherian rings in algebra.

Aliases (1)

Alexandrov topology ×1

Statements (41)

Predicate	Object
instanceOf	mathematical concept → topological notion →
analogy	Noetherian modules with descending chain condition on submodules → Noetherian posets with descending chain condition on subsets →
arisesFrom	spectrum of a Noetherian ring with the Zariski topology →
characterizedBy	every open cover of any open subset has a finite subcover → every subset is compact if and only if it is closed →
context	Zariski topology → general topology →
definition	a topological space in which every descending chain of closed subsets stabilizes →
equivalentDefinition	a topological space in which every ascending chain of open subsets stabilizes → a topological space in which every nonempty collection of closed subsets has a minimal element under inclusion → a topological space in which every open subset is quasi-compact → a topological space in which every subset is compact if and only if it is closed →
example	Spec(R) with the Zariski topology for a Noetherian ring R → a finite T0 space → a finite discrete space →
field	topology →
generalizationOf	finite topological space →
hasFinitenessCondition	ascending chain condition on open sets → descending chain condition on closed sets →
implies	every closed subset is quasi-compact → every open subset is quasi-compact → every subset is a finite union of locally closed subsets →
namedAfter	Emmy Noether →
nonExample	any infinite discrete space → the real line with the usual topology →
property	Noetherian spaces satisfy the ascending chain condition on open sets → Noetherian spaces satisfy the descending chain condition on closed sets → a Noetherian space is quasi-compact → a Noetherian space need not be Hausdorff → every closed subset is a Noetherian space with the subspace topology → every continuous image of a Noetherian space is Noetherian → finite topological spaces are Noetherian → in a Noetherian space every nonempty closed subset has an irreducible component → in a Noetherian space every open subset is a finite union of irreducible open subsets → in a Noetherian space every subset is a finite union of irreducible closed subsets →
relatedTo	Noetherian rings → surface form: "Noetherian ring"
usedIn	algebraic geometry → commutative algebra → scheme theory →

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Noetherian induction → appliesTo → Noetherian space →

Emmy Noether → hasHonorificName → Noetherian space →

Pavel Alexandrov → notableFor → Noetherian space →

this entity surface form: "Alexandrov topology"