Alexandrov–Hausdorff theorem
E174093
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Alexandrov–Hausdorff theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1509445 — 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: Alexandrov–Hausdorff theorem Context triple: [Pavel Alexandrov, notableFor, Alexandrov–Hausdorff theorem]
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A.
Alexandrov compactification
The Alexandrov compactification is a topological construction that adds a single “point at infinity” to a non-compact space to make it compact.
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B.
Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
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C.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
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D.
Carathéodory’s extension theorem
Carathéodory’s extension theorem is a fundamental result in measure theory that guarantees a unique extension of a pre-measure defined on an algebra of sets to a complete measure on the generated σ-algebra.
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E.
Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
- 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: Alexandrov–Hausdorff theorem Target entity description: 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.
-
A.
Alexandrov compactification
The Alexandrov compactification is a topological construction that adds a single “point at infinity” to a non-compact space to make it compact.
-
B.
Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
-
C.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
-
D.
Carathéodory’s extension theorem
Carathéodory’s extension theorem is a fundamental result in measure theory that guarantees a unique extension of a pre-measure defined on an algebra of sets to a complete measure on the generated σ-algebra.
-
E.
Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in descriptive set theory ⓘ |
| appliesTo | subsets of Polish spaces ⓘ |
| assumes | underlying space is Polish ⓘ |
| characterizes | analytic sets as continuous images of Baire space ⓘ |
| classification | characterization theorem for analytic sets ⓘ |
| concerns |
Baire space
ⓘ
Polish space ⓘ analytic set ⓘ |
| domain |
Polish spaces
ⓘ
separable completely metrizable topological spaces ⓘ |
| field |
descriptive set theory
ⓘ
set theory ⓘ topology ⓘ |
| guarantees | every analytic set is image of a continuous map from a universal Polish space ⓘ |
| historicalPeriod | early 20th century mathematics ⓘ |
| implies |
analytic sets are projections of Borel sets in a product of Polish spaces
ⓘ
every analytic set is a continuous image of a closed subset of Baire space ⓘ |
| importance |
fundamental result in descriptive set theory
ⓘ
key tool in the study of definable sets in Polish spaces ⓘ |
| involves |
Baire space ω^ω
ⓘ
standard Borel spaces ⓘ |
| mathematicsSubjectClassification |
03E15
ⓘ
54H05 ⓘ |
| namedAfter |
Felix Hausdorff
ⓘ
Pavel Alexandrov ⓘ |
| relatedConcept |
standard representation of analytic sets via Baire space
ⓘ
universal Polish space ⓘ |
| relatesTo |
Lusin–Souslin theorem
ⓘ
Souslin operation ⓘ definable sets in Polish spaces ⓘ projective hierarchy ⓘ |
| role |
links descriptive set theory with general topology
ⓘ
provides a structural characterization of analytic sets ⓘ |
| statement |
A subset of a Polish space is analytic if and only if it is the continuous image of Baire space
ⓘ
Every analytic subset of a Polish space is the continuous image of Baire space ⓘ |
| typicalFormulation | For any analytic set A in a Polish space X there exists a continuous f from Baire space to X with f[Baire space] = A ⓘ |
| usedIn |
analysis of measurable and category properties of analytic sets
ⓘ
classification of subsets of Polish spaces by descriptive complexity ⓘ construction of non-Borel analytic sets ⓘ |
| usesConcept |
Borel set
ⓘ
continuous function ⓘ projection of a Borel set ⓘ |
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: Alexandrov–Hausdorff theorem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.