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Smooth partitions subordinate to a countable coordinate cover
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Every countable cover of a smooth manifold by coordinate balls admits a smooth partition of unity subordinate to that cover.
Facts & Assumptions
Given: and a countable coordinate-ball cover of a smooth manifold.
Under , every open cover of a smooth manifold admits a subordinate smooth partition of unity (Smooth partitions of unity exist on manifolds).
Proof
A countable coordinate-ball cover is an open cover.
Apply [L1] to that open cover.
The resulting partition is subordinate to the given countable coordinate-ball cover.
Depends on
Used by
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
- 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)