Suppose we are given a smooth manifold $M$ and, for the sake of simplicity, some compact submanifold $L\subseteq M$ of the same dimension, as well as $f\in C^{\infty}(M,N)$ and some submanifold $V\subseteq J^{k}(M,N)$ such that $(j^{k}f)(M\setminus \mathring{L})\cap V=\emptyset\,$ (as a matter of fact I am only interested $f(M\setminus \mathring{L})$ is actually disjoint from the projection of $V$ onto $N$).
My question is whether for any neighbourhood $\mathcal{W}$ of $f$ in the (strong) Whitney topology it is possible to find some $f'\in \mathcal{W}$ such that $j^{k}f'\pitchfork V$ and $f_{\restriction N\setminus \mathring{L}}=f_{\restriction M\setminus\mathring{L}}$.
Now, if one were working in $C^{r}$ for finite $r$, it seems like it would be able to specify some neighbourhood $\mathcal{W}_{0}$ of $f_{\restriction \mathring{L}}$ in the Whitney topology on $C^{r}(\mathring{L},N)$ such that for any $h\in\mathcal{W}_{0}$ the function $h\cup f_{\restriction N\setminus\mathring{L}}$ is $C^{R}$, so that the problem would be solved by a direct application of Thom's transversality to $C^{\infty}(\mathring{L},N)$. However, for $C^{\infty}$ this fails, since the Whitney $C^{\infty}$ topology is just the direct limit of the $C^{r}$ topologies and thus no open condition can control the convergence of all possible higher order derivatives as we approach $\partial L$.
While doing some reading I came upon P.W. Michor's book "Manifolds of differentiable mappings", where he defines what he calls the $\mathcal{D}$-topology on $C^{\infty}(M,N)$. A basis of open sets is given as follows. For any choice of compact sets $\{K_{n}\}_{n\in\mathbb{N}}$ with $\emptyset=K_{0}$, $K_{n}\subseteq\mathring{K}_{n+1}$ and $\bigcup_{n\in\mathbb{N}}K_{n}=M$, any sequence of natural numbers $(m_{n})_{n\in\mathbb{N}}$ and of open sets $U_{n}\subseteq J^{m_{n}}(M,N)$ there is a basic open set of the $\mathcal{D}$-topology is given by the intersection of all the conditions $\{f\,|\,(j^{m_{n}}f)(M\setminus \mathring{K}_{n})\subseteq U_{n}\}$. He later shows that $C^{\infty}(M,N)$ with the $\mathcal{D}$-topology is Baire and proves that Thom's transversality theorem is satisfied: for any given submanifold $V\subseteq J^{k}(M,N)$ (in fact, more generally, for maps) the set of $g\in C^{\infty}(M,N)$ for which $j^{k}g\pitchfork V$ for any given submanifold $V\subseteq J^{k}(M,N)$ is residual. It would seem as if this solves the problem, since given $\mathcal{W}$ as above any $h$ in a sufficiently small neighbourhood of $f_{\restriction \mathring{L}}$ in the $\mathcal{D}$ topology will certainly satisfy $f_{\restriction M\setminus\mathring{L}}\cup h$ is $\mathcal{W}$.
Is the statement I am trying to prove correct? If so, is this a sound strategy? Is there any quicker way of seeing it? Any results in the literature one could directly cite?
[ I checked this question (in the equivariant setting) before posting. I was not convinced by the answer there, since nowhere in the statement of the Proposition being referenced does it say that the new function has to agree with the old one on $C$. Please, forgive me if I'm making a mistake. ]
