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.
Bounded distance between two maps into a metric space
Definition
Let be a set, let be a metric space, and let .
The maps and are at bounded distance from one another if there is a real such that
In that case one also says that and are at finite distance.
Depends on
Used by
- Coarsely dense subsets, quasi-inverses and quasi-isometries Definition
- The quasi-isometry group of a metric space Definition
- A nonempty metric space of finite diameter has trivial quasi-isometry group Example
- A map at bounded distance from a quasi-isometric embedding is one, with the additive constant enlarged Lemma
- Bounded distance is an equivalence relation and is preserved by pre-composition and by post-composition with a coarse Lipschitz map Lemma
- A quasi-isometric embedding with coarsely dense image has a quasi-inverse quasi-isometric embedding Theorem
Dependency tree · two levels
7 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
- C. Loh, Geometric Group Theory: An Introduction (2015 course version), Section 5.1 (standard reference, not scraped)