On a compact smooth manifold, the Morse functions form an open dense subset in the $C^2$ and hence $C^{\mathrm{\infty }}$ topology

Use the assumed Axiom of Choice to apply the conditional density theorem [L1] (with the closed set empty): given any smooth function and any $C^{\mathrm{\infty }}$ neighbourhood of it, there is a Morse function in that neighbourhood. Since the $C^{\mathrm{\infty }}$ topology is finer than the $C^2$ topology on the same compact manifold, this already implies density in the $C^2$ topology as well.

Let $f$ be Morse. Apply [L2] to choose pairwise disjoint critical neighbourhoods $U_1,\dots ,U_r$ for the critical points of $f$, with the stated persistence and Hessian-gap properties. Apply [L3] to the complement $M\setminus {\bigcup }_i{U}_i$. Then every function $g$ sufficiently close to $f$ in the $C^2$ topology has exactly one critical point in each $U_i$, all those critical points are nondegenerate by [L2], and there are no further critical points outside the $U_i$ by [L3].

Therefore every critical point of such a nearby function $g$ is nondegenerate. Hence $g$ is Morse. This proves that the Morse functions are open in the $C^2$ topology.

Combining steps 1.1 and 2.1 gives that the Morse functions form an open dense subset in the $C^2$ topology. Because the identity map from the $C^{\mathrm{\infty }}$ topology to the $C^2$ topology is continuous on a compact source, the same set is also open and dense in the $C^{\mathrm{\infty }}$ topology.