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accepted Is the space of smooth sections of a vector bundle finitely generated as a $C^\infty$-module?
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comment Is the space of smooth sections of a vector bundle finitely generated as a $C^\infty$-module?
@EricO.Korman: Actually I would be already be satisfied to see an example of a series of vector bundles on manifolds with constant rank but unbounded size of minimal $C^\infty$-generating set.
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comment Is the space of smooth sections of a vector bundle finitely generated as a $C^\infty$-module?
Thank you for the link. Unfortunately, this yields an affirmative answer only in the case that M is compact.