TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The interior is an open smooth n-manifold

Statement

For an $n$ -manifold with boundary, $Int M$ is open and, with restricted charts, is a smooth boundaryless $n$ -manifold.

Facts & Assumptions

Given: A smooth $n$ -manifold $M$ with boundary.

[L1]

For $n > 0$ , interior points have positive last boundary-chart coordinate; for $n = 0$ , every point is interior, and these classifications are intrinsic (Interior and boundary of a manifold with boundary; Smooth invariance of the manifold boundary).

[L2]

Boundary-chart images are relatively open in $H^{n}$ , and their transition maps are smooth (Smooth charts, atlases, and structures with boundary).

Proof

technique · direct

1.1

If $n = 0$ , [L1] gives $Int M = M$ ; it is open, and its original charts have image in $R^{0}$ , so the conclusion follows. Assume $n \geq 1$ . Restrict each boundary chart to the inverse image of ${x^{n} > 0}$ . These sets are open and, by [L1], cover exactly $Int M$ .

givenL1algebra

2.1

In the $n \geq 1$ case, their images are Euclidean-open, and [L2] shows that the old transition maps restrict to ordinary smooth transitions. The restricted charts therefore make $Int M$ a smooth boundaryless $n$ -manifold; step 1.1 already handled $n = 0$ .

givenL2step 1.1∎

Depends on

Used by

Morse-Sard for maps from manifolds with boundary Theorem

Dependency tree · two levels

8 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)