TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-30

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Smooth partitions of unity exist on manifolds

Statement

Every open cover of a smooth manifold admits a smooth partition of unity subordinate to it.

Facts & Assumptions

Given: A smooth manifold $M$ and an open cover $U$ of $M$ .

[L1]

The cover has a countable subordinate cover by relatively compact coordinate balls (Every open cover of a manifold has a countable relatively compact coordinate-ball subcover).

[L2]

Such a countable cover has a countable locally finite shrinking $\overline{W_{k}} \subseteq V_{k} \subseteq U_{n (k)}$ (A countable coordinate-ball cover has a countable locally finite shrinking).

[L3]

For every compact set inside an open set there is a smooth manifold bump equal to $1$ on a neighbourhood of that compact set and supported in the open set (A manifold bump for a compact set inside an open set).

[L4]

A locally finite nonnegative smooth family that is pointwise positive normalizes to a partition of unity (A locally finite positive smooth family normalizes to a partition of unity).

Proof

technique · direct

1.1

Apply [L1] and then [L2] to obtain countably many open sets $W_{k}$ and coordinate balls $V_{k}$ such that $M = ⋃_{k} W_{k}$ , the family $(V_{k})$ is locally finite, and each $V_{k}$ lies in some member $U_{n (k)}$ of $U$ .

L1L2given

2.1

For each $k$ , apply [L3] to the compact set $\overline{W_{k}} \subseteq V_{k}$ to obtain a smooth function $g_{k} : M \to [0, 1]$ that equals $1$ on a neighbourhood of $\overline{W_{k}}$ and is supported in $V_{k}$ . The family $(g_{k})$ is locally finite and pointwise positive because every point lies in some $W_{k}$ .

L3step 1.1choose

3.1

Normalize $(g_{k})$ by [L4]; the resulting family $(ϕ_{k})$ is a smooth partition of unity, and $supp (ϕ_{k}) \subseteq V_{k} \subseteq U_{n (k)}$ for every $k$ .

L4step 2.1

4.1

Hence $(ϕ_{k})$ is subordinate to $U$ in the sense of Smooth partitions of unity subordinate to an open cover.

step 3.1∎

Depends on

Used by

Dependency tree · two levels

16 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

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)