Baire category theorem
E518477
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Baire category theorem canonical | 4 |
How this entity was disambiguated
This entity first appeared as the object of triple T5425760 — 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: Baire category theorem Context triple: [Banach inverse mapping theorem, proofTechnique, Baire category theorem]
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A.
Banach–Steinhaus theorem
The Banach–Steinhaus theorem is a fundamental result in functional analysis that characterizes when a family of continuous linear operators is uniformly bounded, with major implications for the behavior of sequences of operators on Banach spaces.
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B.
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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C.
Banach–Mazur theorem
The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
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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.
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E.
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.
- 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: Baire category theorem Target entity description: 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.
-
A.
Banach–Steinhaus theorem
The Banach–Steinhaus theorem is a fundamental result in functional analysis that characterizes when a family of continuous linear operators is uniformly bounded, with major implications for the behavior of sequences of operators on Banach spaces.
-
B.
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.
-
C.
Banach–Mazur theorem
The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
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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.
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.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
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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.