Hausdorff maximal principle

E608817

mathematical theorem result in order theory set-theoretic principle

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.

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Observed surface forms (1)

Surface form	Occurrences
Zorn's lemma	1

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 ⓘ

Referenced by (2)

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

Felix Hausdorff → notableConcept → Hausdorff maximal principle ⓘ

Cantor–Bernstein–Schröder theorem → relatedTo → Hausdorff maximal principle ⓘ

this entity surface form: Zorn's lemma