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.
Topological manifolds with boundary
Definition
An -dimensional topological manifold with boundary is a Hausdorff, second-countable space for which every has a neighbourhood homeomorphic to a relatively open subset of . Dimension is allowed, using .
Depends on
Used by
- Interior and boundary of a manifold with boundary Definition
- Pointwise Borel nonnegative densities Definition
- Smooth charts, atlases, and structures with boundary Definition
- Smooth cobordism triad for Morse theory Definition
- Smooth foliations tangent to the boundary Definition
- The closed half-space as a manifold with boundary Example
- A curvilinear triangulation gives a finite regular CW complex Lemma
- A nonempty compact connected one-dimensional manifold without boundary is a circle Lemma
- Gluing continuation solutions gives collar neighbourhoods of the broken ends Lemma
- 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
- Empty boundary is equivalent to being boundaryless Proposition
- Countable mayer vietoris open set principle Theorem
- Ordered configuration spaces cover the unordered ones regularly with deck group Sₙ Theorem
- The index-two compactification is a compact one-manifold with boundary Theorem
- Transverse preimages for maps from manifolds with boundary Theorem
Dependency tree · two levels
11 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
- 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)