Alphabeta Math
False statementConstruction: Literature-sourcedVerification: 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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Can a smooth chart turn a boundary point into an interior point?

Statement

False. A smooth boundary-chart transition cannot send a face point to a relative-interior point.

Facts & Assumptions

Given: Two compatible boundary charts whose overlap transition is f:U→V, and a face point p∈U.

[L1]

A smooth diffeomorphism between relatively open half-space sets maps face points to face points (Smooth invariance of the manifold boundary).

Refutation

technique · direct
1.1

Compatibility of the two charts makes f a smooth half-space diffeomorphism.

given
2.1

By [L1], f(p) is a face point, not a relative-interior point. Thus no smooth boundary-chart transition can make the proposed change.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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