Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31
How 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.
Smooth embeddings
Definition
Let be a smooth map. Then is a smooth embedding when
- is injective,
- is an immersion (Immersions, submersions, and constant-rank maps), and
- is a homeomorphism, where carries the subspace topology from (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological, Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).
Depends on
Immersions, submersions, and constant-rank maps
Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological
Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace