I am trying to follow the Thom's Transversality Theorem proof from Golubitsky-Guillemin "Stable mappings and Their Singularities" (theorem 4.9, the jet version). The proof is fairly long and my only problem is how they apply the Parametric Transversality Theorem (corollary 4.7) at the end of the proof for the "residual" part, so I will try to summarize the situation.
(For a quick but self-contained reference on the topic see this thesis. Chapter 3 reviews transversality and jet bundles, theorem 3.4.2 is the jet transversality theorem, the proof here follows closely from the one in the mentioned book, and lemma 3.1.5 is the parametric trasnversality theorem.)
The crux of the argument is to prove that for a given open subset $W_r$ of the regular submanifold $W$ in $J^r(X,Y),$ such that $\overline W_r$ is compact and $\overline W_r \subseteq W,$ and also $\sigma(\overline W_r) \subseteq U,$ $\tau(\overline W_r) \subseteq V,$ where $U, V$ are coordinate nbhds of $X$ and $Y,$ respectively, $\overline U$ is compact, and $\sigma, \tau$ are the source and target projections, then set
$$T_{W_r} = \{f \in C^\infty(X,Y) : j^r f \pitchfork W \text{ on } \overline W_r \}$$
is dense in the Whitney $C^\infty$ (strong) topology.
The overall idea for the proof is as follows: for a given smooth function $f,$ they construct family of functions parametrized by the set $B'$ of all polynomials with degree $\leq r,$ the "polynomial perturbation" $g_p$ of $f,$ $p \in B'.$ We then consider the function $$ \Phi : X \times B' \to J^r(X,Y), \quad \Phi(x,p) = j^r g_p(x),$$ which is smooth. By reducing $B'$ to a clever open set $B$ of polynomials around the zero polynomial, they then proceed to show that $\Phi : X \times B \to J^r(X,Y)$ is transverse to $W$ on $\overline W_r.$ (the proof actually states "transverse on some nbhd of $\overline W_r$", but no such nbhd of $\overline W_r$ is mentioned for the rest of the proof).
To prove this, take an $(x,p) \in X \times B'.$ Either $\Phi(x,p) \notin \overline W_r$ and the transversality condition on $\overline W_r$ is trivially satisfied, or $\Phi(x,p) \in \overline W_r,$ and in this case they show that on a nbhd of $(x,p)$ in $X \times B,$ $\Phi$ is a diffeomorphism, so clearly $\Phi$ is transverse to $W$ on $\overline W_r.$
Now here is my problem/question: Knowing that $\Phi : X \times B \to J^r(X,Y)$ is transverse to $W$ on $\overline W_r$ we can "apply the "Parametric Transversality Theorem," so the set of polynomials $p \in B$ such that $g_p$ is transverse to $W$ on $\overline W_r$ is dense in $B.$" If this is true the rest of the proof follows. However I don't see how we can apply the usual Parametric Transversality Theorem for this situation. As this theorem is usually stated (and also how is stated in this book) we need the transversality condition on all the submanifold $W.$ Here we are only satisfying it on the compact subset $\overline W_r$ (which might not be a submanifold). What if $(x,p)$ is such that $\Phi(x,p) \notin \overline W_r,$ but this point might still be in $W,$ and we don't know if it satisfies the transversality condition as to apply the parametric theorem.
So, how can this be done? Or am I missing something? Is there a "transverse to a closed subset of $W$" version of the parametric theorem?
If we could show that actually $\Phi$ is transverse to a nbhd of $\overline W_r$ in $W$ (it is what is stated but not clear in the text if it is done) than this is an open set of a submanifold, therefore also a submanifold and the parametric theorem could be applied, but I couldn't find any such nbhd.
Also, we only show that $\Phi$ is a local diffeomorphism for points on the domain for which the image lies in $\overline W_r,$ not that $\Phi$ is a local diffeomorphism over it's whole domain, if that would be true then the problem is done but does not seem to be the case...
Anyway, I hope I made my issue clear, the usage of the parametric transversality theorem doesn't seem to be correct as stated, it either needs some "closed subset" version (if such is even possible, I couldn't prove one stated as such), or some more clarification for this "nbhd of $\overline W_r$" is needed.
Any help, hints or references for this question are appreciated. Thanks in advance.