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In much of the Morse Theory literature, it seems that we always assume $M$ is a smooth closed (or compact) manifold so that we get theorems like:

The set of Morse-Smale gradient vector fields on $M$ is "generic."

I would rather $M$ be an open subset of $\mathbb{R}^n$ (which is certainly not a closed manifold). Under what conditions will the above theorem hold if I let $M$ be an open subset of $\mathbb{R}^n$ instead? Is there a reference for something like this?

If $f \colon M \to \mathbb{R}$, I think we need grad $f$ transversally pointing inward on $\partial M$?

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It holds. On closed manifolds one can upgrade the genericity to open and dense, which is stronger. – Thomas Rot Aug 6 '14 at 15:39

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