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

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Smooth partitions of unity exist on manifolds with boundary

Statement

Assume ${A C}_{ω}$ . Every open cover of a smooth manifold with boundary admits a smooth partition of unity subordinate to it.

Facts & Assumptions

Given: ${A C}_{ω}$ , a smooth manifold with boundary, and an open cover.

Proof

technique · direct

1.1

In half-space charts, restrict the Euclidean bumps used in the boundaryless construction; their supports and local finiteness survive restriction.

given

2.1

The same countable locally finite shrinking and normalization construction produces a positive locally finite family summing to one. Thus it is subordinate to the prescribed cover, with the same ${A C}_{ω}$ -sufficient choice bound as the precursor construction.

step 1.1∎

Depends on

Used by

A global inward-pointing boundary vector field exists Theorem
Collar neighborhood theorem Theorem
Orientability is equivalent to a nowhere-vanishing top form Theorem

Dependency tree · two levels

12 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)