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

To prove density, let $\mathcal{U}$ be a $C^{\mathrm{\infty }}$ neighbourhood of an arbitrary smooth function $f$. Using the assumed Axiom of Choice to apply [L1], choose a Morse function $h\in \mathcal{U}$. Because $\mathcal{U}$ is itself a $C^{\mathrm{\infty }}$ neighbourhood of the compact Morse function $h$, [L2] yields a function $g\in \mathcal{U}$ with the same critical points and Hessians as $h$, but with pairwise distinct critical values. By [F1], this $g$ is excellent Morse.

Let $f$ be excellent Morse. By [L3], choose pairwise disjoint critical neighbourhoods $U_1,\dots ,U_r$ around the critical points $p_1,\dots ,p_r$ of $f$ such that every sufficiently small $C^2$ perturbation has exactly one nondegenerate critical point in each $U_i$ and none outside ${\bigcup }_i{U}_i$. Because the critical values $f(p_i)$ are pairwise distinct, choose pairwise disjoint open intervals $I_i$ with $f(p_i)\in I_i$ and $f(U_i)\subseteq I_i$.

If $g$ is sufficiently close to $f$ in the $C^2$ topology, then step 1.2 gives exactly one critical point $q_i$ of $g$ in each $U_i$ and no others. The $C^0$ closeness keeps $g(q_i)$ inside the same interval $I_i$, so the critical values of $g$ are pairwise distinct because the intervals are disjoint. Since each $q_i$ is nondegenerate, [F1] shows that $g$ is excellent Morse.

Step 1.1 gives density in the $C^{\mathrm{\infty }}$ topology, hence also in the coarser $C^2$ topology, and step 2.1 gives openness in the $C^2$ topology. Every $C^2$-open subset is also $C^{\mathrm{\infty }}$-open on a compact manifold, so the same set of excellent Morse functions is open in the $C^{\mathrm{\infty }}$ topology as well. Thus the excellent Morse functions are open dense in both topologies.