In Audin Damian's proof of the Invariance of Morse Homology from vector field and Morse function, p.71, there's a proposition, where the Smale property is that the stable and unstable manifolds of the critical points of a Morse function are all transverse (copying the paragraph before for context):
The language of the paragraph hints that the author does not seem to be giving a lot of details in here (also the white square after the proposition is because they actually did not provide a proof). Even though this statement is credible it does not seem straightforward to prove at all. Would someone have a suggestion on how to start on something like this?
I thought of this proposition in the following form: the set of pseudo-gradient vector fields adapted to a fixed Morse function $f$ with the Smale property and a given set of trajectories is open in the set $\mathfrak{X}(V)$ of vector fields metrized with the $C^1$ norm.
This notion of $C^1$ proximity is explained in Audin and Damian's notion of $C^1$ proximity