Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-09-07
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.
Euclidean upper half-space and its boundary
Definition
For , put , with the subspace topology, and put . Put and .
Used by
- Smooth functions on relatively open half-space sets Definition
- Topological manifolds with boundary Definition
- A circle is the boundary of a disk Example
- Signed points give the oriented zero-bordism invariant Example
- The closed half-space as a manifold with boundary Example
- The pair of pants is a cobordism realizing addition of circles Example
- Two unoriented points bound an interval Example
- Relatively compact coordinate balls and half-balls form a boundary-manifold basis Lemma
- The closed disk D² is a connected Hausdorff topological 2-manifold with boundary Lemma
- Transverse preimages carry the pulled-back normal structure Proposition
- Zero-dimensional bordism groups Proposition
- Ordered configuration spaces cover the unordered ones regularly with deck group Sₙ Theorem
- Transverse preimages for maps from manifolds with boundary Theorem
Dependency tree · 0 levels
Nothing. This result depends on no other item in the library.
Sources
- Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
- Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)