Lusin–Souslin theorem
E681624
The Lusin–Souslin theorem is a fundamental result in descriptive set theory stating that the continuous injective image of a Borel set in a Polish space is again a Borel set.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Lusin–Souslin theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7685000 — 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: Lusin–Souslin theorem Context triple: [Alexandrov–Hausdorff theorem, relatesTo, Lusin–Souslin theorem]
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A.
Baire category 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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B.
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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C.
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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D.
Cantor’s theorem
Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
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E.
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.
- 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: Lusin–Souslin theorem Target entity description: The Lusin–Souslin theorem is a fundamental result in descriptive set theory stating that the continuous injective image of a Borel set in a Polish space is again a Borel set.
-
A.
Baire category 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.
-
B.
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.
-
C.
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.
-
D.
Cantor’s theorem
Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
-
E.
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.
- F. None of above. chosen
Statements (35)
| Predicate | Object |
|---|---|
| instanceOf | mathematical theorem ⓘ |
| appliesTo |
Borel sets
NERFINISHED
ⓘ
Polish spaces ⓘ continuous injective functions ⓘ |
| codomainCondition | codomain is a Polish space ⓘ |
| concerns | structure of Borel sets under continuous injections ⓘ |
| conclusion | image set is Borel in the codomain Polish space ⓘ |
| context | classical descriptive set theory ⓘ |
| domainCondition | domain is a Borel subset of a Polish space ⓘ |
| ensures | injective Borel measurable images under continuous maps remain Borel ⓘ |
| field |
descriptive set theory
ⓘ
mathematical logic ⓘ set theory ⓘ |
| formalizes | stability of Borel sets under continuous injective images in Polish spaces ⓘ |
| hasConsequence |
continuous bijections between Borel subsets of Polish spaces are Borel isomorphisms
ⓘ
graphs of Borel measurable injective functions between Polish spaces are Borel ⓘ |
| historicalPeriod | early 20th century mathematics ⓘ |
| implies | continuous injective images of Borel sets preserve Borel measurability ⓘ |
| language | mathematical analysis ⓘ |
| mapCondition | map is continuous and injective ⓘ |
| namedAfter |
Mikhail Souslin
NERFINISHED
ⓘ
Nikolai Luzin NERFINISHED ⓘ |
| relatedTo |
Borel hierarchy
NERFINISHED
ⓘ
Luzin hierarchy NERFINISHED ⓘ Souslin operation NERFINISHED ⓘ Souslin theorem NERFINISHED ⓘ analytic sets ⓘ |
| statement | The continuous injective image of a Borel set in a Polish space is a Borel set. ⓘ |
| strengthens | basic facts about continuity and Borel measurability on Polish spaces ⓘ |
| typeOfResult | regularity theorem for Borel sets ⓘ |
| usedIn |
Borel equivalence relations
ⓘ
classification of Borel sets ⓘ descriptive set theory of Polish spaces ⓘ measurable dynamics on Polish spaces ⓘ theory of Borel isomorphisms ⓘ |
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
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Input
Subject: Lusin–Souslin theorem Description of subject: The Lusin–Souslin theorem is a fundamental result in descriptive set theory stating that the continuous injective image of a Borel set in a Polish space is again a Borel set.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.