Triple

T14704225
Position Surface form Disambiguated ID Type / Status
Subject Émile Borel E345382 entity
Predicate notableWork P4 FINISHED
Object Borel–Lebesgue theorem
The Borel–Lebesgue theorem is a fundamental result in real analysis and topology that characterizes compact subsets of Euclidean space via the property that every open cover admits a finite subcover.
E1116059 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Borel–Lebesgue theorem | Statement: [Émile Borel, notableWork, Borel–Lebesgue theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Borel–Lebesgue theorem
Context triple: [Émile Borel, notableWork, Borel–Lebesgue theorem]
  • A. Arzelà–Ascoli theorem
    The Arzelà–Ascoli theorem is a fundamental result in analysis that characterizes the relative compactness of families of functions via uniform boundedness and equicontinuity.
  • B. Bolzano–Weierstrass theorem
    The Bolzano–Weierstrass theorem is a fundamental result in real analysis stating that every bounded infinite sequence in ℝⁿ has a convergent subsequence.
  • C. 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.
  • D. 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.
  • E. Krein–Milman theorem
    The Krein–Milman theorem is a fundamental result in functional analysis and convex geometry stating that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Borel–Lebesgue theorem
Triple: [Émile Borel, notableWork, Borel–Lebesgue theorem]
Generated description
The Borel–Lebesgue theorem is a fundamental result in real analysis and topology that characterizes compact subsets of Euclidean space via the property that every open cover admits a finite subcover.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Borel–Lebesgue theorem
Target entity description: The Borel–Lebesgue theorem is a fundamental result in real analysis and topology that characterizes compact subsets of Euclidean space via the property that every open cover admits a finite subcover.
  • A. Arzelà–Ascoli theorem
    The Arzelà–Ascoli theorem is a fundamental result in analysis that characterizes the relative compactness of families of functions via uniform boundedness and equicontinuity.
  • B. Bolzano–Weierstrass theorem
    The Bolzano–Weierstrass theorem is a fundamental result in real analysis stating that every bounded infinite sequence in ℝⁿ has a convergent subsequence.
  • C. 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.
  • D. 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.
  • E. Krein–Milman theorem
    The Krein–Milman theorem is a fundamental result in functional analysis and convex geometry stating that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.
  • F. None of above. chosen

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69d822e4a8c08190a155df736bb7bc13 completed April 9, 2026, 10:06 p.m.
NER Named-entity recognition batch_69deb6071e5c8190bb5509c859135c2d completed April 14, 2026, 9:47 p.m.
NED1 Entity disambiguation (via context triple) batch_69fdf087ce8c819081a7186df67bcf1f completed May 8, 2026, 2:17 p.m.
NEDg Description generation batch_69fdf2a63cc88190b3670378c54c96b6 completed May 8, 2026, 2:26 p.m.
NED2 Entity disambiguation (via description) batch_69fdf31fcb4081908a88cf4d4c5ddced completed May 8, 2026, 2:28 p.m.
Created at: April 10, 2026, 1:28 a.m.