Triple

T16474847
Position Surface form Disambiguated ID Type / Status
Subject Tychonoff theorem for products of compact spaces E400161 entity
Predicate generalizes P2372 FINISHED
Object Heine–Borel theorem for products of closed bounded intervals in R E1116059 NE FINISHED

How this triple was built (2 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: Heine–Borel theorem for products of closed bounded intervals in R | Statement: [Tychonoff theorem for products of compact spaces, generalizes, Heine–Borel theorem for products of closed bounded intervals in R]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07 
Target entity: Heine–Borel theorem for products of closed bounded intervals in R
Context triple: [Tychonoff theorem for products of compact spaces, generalizes, Heine–Borel theorem for products of closed bounded intervals in R]
  • A. Borel–Lebesgue theorem chosen
    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.
  • B. Tychonoff theorem for products of compact spaces
    The Tychonoff theorem for products of compact spaces is a fundamental result in topology stating that any product of compact topological spaces is compact, a statement that is equivalent in strength to the axiom of choice.
  • C. 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.
  • D. 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.
  • E. 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.
  • F. None of above.
  • G. Unsure - the case is ambiguous/there is not enough information to decide.

Provenance (3 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_69d883813098819084f5409539723b59 completed April 10, 2026, 4:58 a.m.
NER Named-entity recognition batch_69e32dd32e048190a9eadd32d6b9374c completed April 18, 2026, 7:08 a.m.
NED1 Entity disambiguation (via context triple) batch_6a004f5f238881909b5f2fb41da3f932 completed May 10, 2026, 9:26 a.m.
Created at: April 10, 2026, 5:13 a.m.