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.
Universal net: eventually in every subset or eventually in its complement
Definition
A net is universal if, for every subset , it is eventually in or eventually in .
The two alternatives cannot both occur: directedness would give an index after both thresholds, whose value would belong to the empty intersection .
Depends on
Used by
- Assuming the ultrafilter lemma, every net has a universal subnet Lemma
- Every cluster point of a universal net is a limit of that net Lemma
- The image of a universal net under any map is universal, and a continuous map preserves its limits Lemma
- A net is universal exactly when its tail filter is an ultrafilter, and the canonical net of an ultrafilter is universal Theorem
Dependency tree · two levels
2 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
- WVU Math 581 Topology I (standard reference, not scraped)
- Net (mathematics) (Wikipedia) (standard reference, not scraped)