Hausdorff maximal principle
E608817
The Hausdorff maximal principle is a foundational result in set theory and order theory stating that every partially ordered set contains a maximal totally ordered subset (a maximal chain), and it is equivalent to the axiom of choice.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hausdorff maximal principle canonical | 1 |
| Zorn's lemma | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6660393 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Hausdorff maximal principle Context triple: [Felix Hausdorff, notableConcept, Hausdorff maximal principle]
-
A.
Hahn–Banach theorem
The Hahn–Banach theorem is a fundamental result in functional analysis that guarantees the extension of bounded linear functionals from a subspace to the whole space without increasing their norm.
-
B.
Krein–Milman theorem
The Krein–Milman theorem is a fundamental result in functional analysis and convex geometry stating that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.
-
C.
Banach–Alaoglu theorem
The Banach–Alaoglu theorem is a fundamental result in functional analysis stating that the closed unit ball in the dual of a normed space is compact in the weak-* topology.
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D.
Alexandrov–Hausdorff theorem
The Alexandrov–Hausdorff theorem is a result in descriptive set theory that characterizes analytic sets as continuous images of Baire space, playing a key role in the study of definable sets in Polish spaces.
-
E.
Cantor–Bernstein–Schröder theorem
The Cantor–Bernstein–Schröder theorem is a fundamental result in set theory stating that if each of two sets can be injected into the other, then there exists a bijection between them, so the sets have the same cardinality.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Hausdorff maximal principle Target entity description: The Hausdorff maximal principle is a foundational result in set theory and order theory stating that every partially ordered set contains a maximal totally ordered subset (a maximal chain), and it is equivalent to the axiom of choice.
-
A.
Hahn–Banach theorem
The Hahn–Banach theorem is a fundamental result in functional analysis that guarantees the extension of bounded linear functionals from a subspace to the whole space without increasing their norm.
-
B.
Krein–Milman theorem
The Krein–Milman theorem is a fundamental result in functional analysis and convex geometry stating that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.
-
C.
Banach–Alaoglu theorem
The Banach–Alaoglu theorem is a fundamental result in functional analysis stating that the closed unit ball in the dual of a normed space is compact in the weak-* topology.
-
D.
Alexandrov–Hausdorff theorem
The Alexandrov–Hausdorff theorem is a result in descriptive set theory that characterizes analytic sets as continuous images of Baire space, playing a key role in the study of definable sets in Polish spaces.
-
E.
Cantor–Bernstein–Schröder theorem
The Cantor–Bernstein–Schröder theorem is a fundamental result in set theory stating that if each of two sets can be injected into the other, then there exists a bijection between them, so the sets have the same cardinality.
- F. None of above. chosen
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in order theory ⓘ set-theoretic principle ⓘ |
| appliesTo | partially ordered sets ⓘ |
| assumes | every chain can be extended to a maximal chain ⓘ |
| category | axiom of choice equivalents ⓘ |
| concerns |
chains in posets
ⓘ
maximal chains ⓘ total orders ⓘ |
| context | Zermelo–Fraenkel set theory without choice NERFINISHED ⓘ |
| equivalentTo |
Zorn's lemma
NERFINISHED
ⓘ
axiom of choice ⓘ well-ordering theorem NERFINISHED ⓘ |
| expressedIn | first-order set theory ⓘ |
| field |
order theory
ⓘ
set theory ⓘ |
| guaranteesExistenceOf |
maximal chain in a poset
ⓘ
maximal totally ordered subset ⓘ |
| holdsIn | any poset ⓘ |
| implies |
Zorn's lemma
NERFINISHED
ⓘ
axiom of choice NERFINISHED ⓘ |
| language | mathematical logic ⓘ |
| logicalStrength | equivalent to axiom of choice over ZF ⓘ |
| namedAfter | Felix Hausdorff NERFINISHED ⓘ |
| relatedConcept |
Zorn's lemma
NERFINISHED
ⓘ
chain ⓘ maximal element ⓘ well-ordering theorem NERFINISHED ⓘ |
| requires | nonempty partially ordered set ⓘ |
| statedAs |
Every partially ordered set has a maximal totally ordered subset
ⓘ
Every poset contains a maximal chain ⓘ |
| status | independent of ZF without choice ⓘ |
| typeOf | maximality principle ⓘ |
| usedIn |
algebra
ⓘ
functional analysis ⓘ model theory ⓘ topology ⓘ |
| usedToProve |
Tychonoff theorem (via equivalence with axiom of choice)
ⓘ
existence of bases in vector spaces ⓘ existence of maximal ideals in rings ⓘ |
| yearIntroduced | early 20th century ⓘ |
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: Hausdorff maximal principle Description of subject: The Hausdorff maximal principle is a foundational result in set theory and order theory stating that every partially ordered set contains a maximal totally ordered subset (a maximal chain), and it is equivalent to the axiom of choice.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Zorn's lemma