Baire category theorem

E518477

mathematical theorem

The Baire category theorem is a fundamental result in topology and functional analysis stating that complete metric (or locally compact Hausdorff) spaces cannot be written as countable unions of nowhere dense sets, with powerful consequences for the structure of such spaces.

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All labels observed (1)

Label	Occurrences
Baire category theorem canonical	4

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ
appearsIn	René-Louis Baire's doctoral thesis ⓘ
appliesTo	Polish spaces ⓘ complete metric spaces ⓘ locally compact Hausdorff spaces ⓘ
characterizes	Baire spaces as spaces where countable intersections of dense open sets are dense ⓘ
concerns	Baire spaces NERFINISHED ⓘ comeagre sets ⓘ complete metric spaces ⓘ locally compact Hausdorff spaces ⓘ meagre sets ⓘ nowhere dense sets ⓘ
contrastsWith	Lebesgue measure theory NERFINISHED ⓘ
field	functional analysis ⓘ topology ⓘ
formalizes	notion of generic properties in topology ⓘ
hasConsequence	Closed graph theorem NERFINISHED ⓘ Open mapping theorem NERFINISHED ⓘ Uniform boundedness principle NERFINISHED ⓘ existence of continuous nowhere differentiable functions ⓘ generic continuity properties of pointwise limits of functions ⓘ generic properties in function spaces ⓘ
historicalPeriod	early 20th century mathematics ⓘ
implies	Complete metric spaces are of second category in themselves ⓘ In a Baire space the intersection of countably many dense open sets is dense ⓘ In a Baire space the union of countably many nowhere dense sets has empty interior ⓘ Locally compact Hausdorff spaces are of second category in themselves ⓘ
introducedBy	René-Louis Baire NERFINISHED ⓘ
isToolFor	proving existence of discontinuous linear functionals ⓘ proving typical behavior of continuous functions on intervals ⓘ
logicalStrength	equivalent to certain forms of the axiom of choice in set theory (in some formulations) ⓘ
namedAfter	René-Louis Baire NERFINISHED ⓘ
relatedTo	Baire space (topology) NERFINISHED ⓘ Banach–Mazur game NERFINISHED ⓘ Polish spaces ⓘ category (topology) ⓘ measure-category duality ⓘ
states	A nonempty complete metric space cannot be expressed as a countable union of nowhere dense subsets ⓘ A nonempty locally compact Hausdorff space cannot be expressed as a countable union of nowhere dense subsets ⓘ Every complete metric space is a Baire space ⓘ Every locally compact Hausdorff space is a Baire space ⓘ
usedIn	Banach space theory ⓘ descriptive set theory ⓘ functional analysis ⓘ operator theory ⓘ topological dynamics ⓘ
yearProposed	1899 ⓘ

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Input

Subject: Baire category theorem
Description of subject: The Baire category theorem is a fundamental result in topology and functional analysis stating that complete metric (or locally compact Hausdorff) spaces cannot be written as countable unions of nowhere dense sets, with powerful consequences for the structure of such spaces.

Referenced by (4)

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

Banach inverse mapping theorem → proofTechnique → Baire category theorem ⓘ

Banach–Steinhaus theorem → requires → Baire category theorem ⓘ

Steinhaus theorem → relatedTo → Baire category theorem ⓘ

Banach–Mazur game → relatedTo → Baire category theorem ⓘ