Recently I have been reading on Morse Homology. Suppose we have a compact manifold $M$ and a smooth function $f:M \rightarrow \mathbb{R}$ and a Morse vector field $X$ such that we can do Morse Homology on the pair $(X,f)$. Later on we will need the Palais-Smale condition and for that we will need to perturb the vector field slightly. I have seen several examples where the specific vector field is constructed, my question is if we can do this in a more abstract setting similar to what happens in Floer theory. In Floer theory, one finds using the Sard-Smale theorem that there exists a residual set of almost complex structures $\mathcal{J}$ such that the Floer equation is transverse to the zero section. I was wondering if we could somehow get a similar result here. If we can prove that there exists a residual set of vector fields $\mathcal{X}$ such that the Palais-Smale condition is satisfied using analogous methods to what is done in Floer theory, i.e., the Sard-Smale theorem? Does anyone know of a referece where I could find this type of idea ?
Any insight is appreciated, thanks in advance.
