I am studying at the moment Thom's transversality theorem for compact and smooth manifolds, which goes like this:
Let $M$ be a compact, smooth manifold. Then the set of Morse functions is a dense and open subset of $\mathcal{C}^\infty(M; \mathbb{R})$ with the $\mathcal{C}^\infty$ topology.
I am following as a reference the book "Transversalité, Courants et Théorie de Morse" by François Laudenbach. I was wondering what one can say when the compactness condition is dropped: in the book it appears to be a fundamental condition already for the construction of Morse functions (I'm thinking in particular about the application of Whitney's embedding theorem to the manifold).
Are there any papers that you know of addressing this issue? Is there, in fact, any interest about it? I know nothing about Morse Theory outside the compact case, and it might very well be that getting a density result for Morse function on non-compact manifolds does not have practical application.