Q:

Prove that regular values are dense (Brown-Sard Theorem), is true by showing it is true for an open subset.

I don't believe this is necessarily related to cohomology. This could be something about manifolds in general.
My book is From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave. 

I. The claim appears to be that if the Brown-Sard Theorem is true for some open subset of $M$ that is diffeomorphic to $\mathbb R^{\dim(M)}$, then the Brown-Sard Theorem is true for the whole of $M$. I guess the $\mathbb R^n$ should be $\mathbb R^m$ instead.
Question: Is there some general idea behind this claim to prove this claim succinctly?
II. I know you can't just spoonfeed me the answer, and I have 2 ways of proving this, but both of them are very long. I hope this is enough effort so that I may be provided a general idea behind the claim or if there is none, to be provided an idea of how to prove this claim.

Way 1. The first way involves guessing that

Step 1.1. if the Brown-Sard Theorem is true for some open subset of $M$ that is diffeomorphic to $\mathbb R^{\dim(M)}$, then the Brown-Sard Theorem is true for every open subset of $M$ that is diffeomorphic to $\mathbb R^{\dim(M)}$.
Step 1.2. if the Brown-Sard Theorem is true for every open subset of $M$ that is diffeomorphic to $\mathbb R^{\dim(M)}$, then the Brown-Sard Theorem is true for the whole of $M$.
In Step 1.1 I consider the diffeomorphism between $V$ and $W$, open sets in $M$ that are both diffeomorphic to $\mathbb R^{\dim(M)}$ and hope that $f^{-1}(V)$ and $f^{-1}(W)$ are diffeomorphic to $\mathbb R^{\dim(N)}$ and thus each other or are at least diffeomorphic to each other anyway. In Step 1.2, I use that the closure of unions contains the union of closures.

Way 2. Here, I prove a contrapositive. If there is some neighborhood $V_p$ of some $p \in M$ such that $V_p  \cap \{\text{regular values of } \ f: N \to M \} = \emptyset$, then for the neighborhood $U_p$ of $p$ that is diffeomorphic to $\mathbb R^n$ and thus diffeomorphic to $W$, we have that there is a neighborhood $X_q$ of $q = \varphi(p)$, where $\varphi$ is the diffeomorphism from $U_p$ to $W$, such that $X_q \cap \{\text{regular values of } \ g: f^{-1}(W) \to W \} = \emptyset$, where

$g = \tilde{h}: f^{-1}(W) \to W$ is the induced map of $h = f \circ \iota = f|_{f^{-1}(W)} : f^{-1}(W) \to M$, $\iota: f^{-1}(W) \to N$ where $\tilde{h}(a) = h(a)$ for all $a \in f^{-1}(W)$, and
$\tilde{h}$ is smooth because $f \circ \iota$ is smooth and $W$ is a regular submanifold of $M$

$W$ is a regular submanifold of $M$ because $W$ is open in $M$.
$f \circ \iota$ is smooth because $\iota$ is smooth because $f^{-1}(W)$ is a regular submanifold of $N$ because $f^{-1}(W)$ is open in $N$.

Specifically, the $X_q$ is I think $\varphi(U_p \cap V_p)$ or something. 

III. Given the length of and possible errors in my 2 aforementioned ways, I think I'm missing something obvious.
This seems to be related to

how an earlier lemma was proved. It seems that

Step 1 was that the lemma was true for $M = \mathbb R^{\dim(M)}$ and then
Step 2 was that if Step 1 holds, then the lemma is true for every open subset of $M$ diffeomorphic to $\mathbb R^{\dim(M)}$ and then
Step 3 was that if the conclusion of Step 2 holds, then the lemma is true for the whole of $M$.

I think the idea of replacing $M$ by $W$, in the Brown-Sard Theorem (Theorem 11.5), is similar to the above Step 3.

how a proposition in An Introduction to Manifolds by Loring W. Tu is proved (Proof. Proposition.). The proof involves "open condition" or "Open property", and the proposition is about sufficient conditions for an immersion or submersion to have constant rank.

I suspect the general idea is that to prove something is true for a manifold, we can prove it is true for some or for every open subset of the manifold that is diffeomorphic to $\mathbb R^{\dim(\text{the manifold})}$

A:

They're not claiming that the theorem being true for some open subset that's diffeomorphic to $\mathbb R^m$ implies that it's true for $M$. What's closer to the truth is that if the theorem is true for every open subset that's diffeomorphic to $\mathbb R^m$, then it's true for $M$. 
Actually, if you read further down the page, you'll see that what they're actually proving is that something stronger is true for every open subset diffeomorphic to $\mathbb R^m$: If $U\subseteq M$ is such a subset, then almost all points in the codomain are regular values of $f|_U$, or equivalently, the set of singular values of $f|_U$ has measure zero. (It follows from this that the set of regular values of $f|_U$ is dense, but it's stronger than that.) Then they explain that you can cover $M$ with countably many such subsets, and use the fact that the union of countably many sets of measure zero again has measure zero.

