Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The compact Hausdorff product theorem uses the ultrafilter lemma, while the published arbitrary compact product theorem assumes the full Axiom of Choice
The proof of Assuming the ultrafilter lemma, an arbitrary product of compact Hausdorff spaces is compact spends the ultrafilter lemma at the universal-subnet step. The published Tychonoff's theorem: an arbitrary product of compact spaces is compact in the product topology, assuming the Axiom of Choice asserts compactness for arbitrary compact factors under the full Axiom of Choice (The Axiom of Choice). These are distinct stated hypotheses; this page makes no claim about their exact relative strength.
Depends on
- Assuming the ultrafilter lemma, an arbitrary product of compact Hausdorff spaces is compact
- Tychonoff's theorem: an arbitrary product of compact spaces is compact in the product topology, assuming the Axiom of Choice
- The ultrafilter lemma, from the Axiom of Choice: every filter extends to an ultrafilter
- The Axiom of Choice
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
25 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Ultrafilter lemma (Wikipedia) (standard reference, not scraped)
- Tychonoff's theorem (Wikipedia) (standard reference, not scraped)
- Boolean prime ideal theorem (Wikipedia) (standard reference, not scraped)