DefinitionDefinition: 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.

Interior and boundary of a manifold with boundary

Definition

Let $p$ lie in an $n$ -manifold with boundary. If $n > 0$ , call $p$ a provisional boundary point if a boundary-chart image has last coordinate $0$ , and an interior point if it has last coordinate $> 0$ . If $n = 0$ , declare every point interior and no point a provisional boundary point. Write these sets as $\partial M$ and $Int M$ ; chart independence is proved below.

Depends on

Used by

Boundary-defining functions Definition
Empty boundary is equivalent to being boundaryless Proposition
Smooth invariance of the manifold boundary Theorem
The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold Theorem
The interior is an open smooth n-manifold 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

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)