Let $G$ be a Lie group acting properly on a smooth manifold $M$. The (non-equivariant) definition of a Morse function does not carry over to equivariant functions $M \rightarrow \mathbb{R}$ (where $\mathbb{R}$ has the trivial action) since critical points are no longer isolated: Every orbit of a critical point will consist entirely of critical points.
However, there is an adaption via Morse-Bott functions $f: M \rightarrow \mathbb{R}$ whose set of critical points is a disjoint union of submanifolds, so called critical submanifolds. Instead of demanding the Hessian at a critical point to be non-degenerate, one requires the Hessian to be non-degenerate on the normal bundle of these submanifolds. We call such submanifolds non-degenerate.
Let us say that a $G$-equivariant map $f : M \rightarrow \mathbb{R}$ is a $G$-equivariant Morse function if it is a Morse-Bott function whose critical submanifolds are non-degenerate orbits.*
If I understood it correctly, in the late 60's Wasserman proved the following (see also lemma 4.8 here).
Let $G$ be a compact Lie group. The set of $G$-equivariant Morse functions is dense in the space of all smooth $G$-equivariant maps $C^{\infty, G}(M, \mathbb{R})$ (equipped with the subspace topology where $C^\infty(M,\mathbb{R})$ has the strong topology).
This result horribly fails when $G$ is non-compact. Indeed, in this paper Illman and Kankaanrinta prove that if $G$ is a non-compact Lie group acting properly on $M$; and $N$ is any $G$-manifold, then the topology on $C^{\infty, G}(M, N)$ induced from the strong topology on $C^{\infty}(M, N)$ is discrete! In their paper, they introduce the so called "strong-weak" topology which sits inbetween the weak and the strong topology on $C^\infty(M,N)$. In particular, if $G$ is compact or acts trivial the strong-weak topology is the strong topology, whereas it is the weak topology if $G$ acts cocompactly.
The strong-weak topology seems to be very adequate for the study of equivariant maps, for instance the authors prove that equivariant proper embeddings between proper $G$-manifolds are open in $C^{\infty, G}(M,N)$ for the strong-weak topology. Similar results can be found in the linked paper.
Of course now I wonder:
Does Wasserman's result remain true in the non-compact case if we consider proper Lie group actions and equip $C^{\infty, G}(M, \mathbb{R})$ with the strong-weak topology?
*(in Wasserman's paper these are precisely the functions in a space called $\mathfrak{M}(M,M)$. His definition of Morse function makes a further technical assumption which I do not need).
