Noetherian space
E29919
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Noetherian space canonical | 2 |
| Alexandrov topology | 1 |
| Noetherian scheme | 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Noetherian space Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Alexandrov topology