Let $M$ be a smooth $k$-manifold with the smooth structure defined in the usual way by an atlas. The weak Whitney embedding theorem says that we may embed $M$ in $\Bbb{R}^{2k+1}$.
In Differential Topology, Guillemin and Pollack prove that if we are given an embedding $M\hookrightarrow \Bbb{R}^N$, then we can inductively reduce the dimension to obtain a (proper) embedding $M\hookrightarrow \Bbb{R}^{2k+1}$.
Thus the natural question is, is there a relatively short proof that a manifold may be embedded in some $\Bbb{R}^N$ for $N$ sufficiently large?
The motivation here is that we can combine this with Guillemin and Pollack's proof to get a fairly short and elegant proof of the first Whitney embedding theorem. There are of course direct proofs of the Whitney embedding theorem that yield such a result, but that's beside the point for this question.
If $M$ is compact:
If $M$ is compact, then it's easy to produce such an embedding, simply cover $M$ by finitely many coordinate charts $(U_i,\phi_i)$, choose a partition of unity $\{\rho_i\}$ subordinate to the $U_i$. Then define $\tilde{\phi_i} = (\rho_i(x)\phi_i(x),\rho_i(x))$ and define the embedding to be $\iota = (\tilde{\phi_i})$. When $\rho_i\ne 0$, $\tilde{\phi_i}$ is an injective immersion, so $\iota$ is locally an embedding, and $\iota$ is injective, since if $\iota(x)=\iota(y)$, then $\rho_i(x)=\rho_i(y)\ne 0$ for some $i$, and then $\phi_i(x)=\phi_i(y)$, so $x=y$. Thus $\iota$ is an injective immersion, and therefore an embedding, since $M$ is compact.
The noncompact case:
Guillemin and Pollack also prove that we can turn an injective immersion into a proper injective immersion (and thus an embedding). Therefore it suffices to produce an injective immersion $M\to \Bbb{R}^N$.
Let's try to adapt the proof in the compact case.
Suppose we can find a finite set of collections $C_i$, $1\le i \le r$ of coordinate open sets $C_i=\{(U_{ij},\phi_{ij}) : j\in \Bbb{N}\}$ with the following properties:
- $\{U_{ij}: 1\le i\le r, j\in \Bbb{N}\}$ is a cover of $M$, and
- $U_{ij}\cap U_{ij'} = \varnothing$, when $j\ne j'$.
Then let $$V_i = \bigcup_{j\in\Bbb{N}} U_{ij},$$ and define $\phi_i : V_i\to \Bbb{R}^{k+1}$ by $\phi_i(x) = (\phi_{ij}(x),j)$, where $x\in U_{ij}$. $\phi_i$ is then an injective immersion from $V_i$ to $\Bbb{R}^{k+1}$. Then we can repeat the proof in the compact case to combine the $\phi_i$s into an injective immersion of all of $M$.
Specific question on this potential method
Can we always produce an open cover with the desired properties? If yes, does my argument work, or have I overlooked something?
Thoughts on this last question
Some googling suggests that the existence of such a cover follows from Ostrand's theorem on colored dimension, according to Mariano Suárez-Álvarez's answer here and wikipedia.
Assuming my argument is correct, this somewhat answers my second question, although given that I can't seem to find a proof of Ostrand's theorem, and I imagine it's a bit involved, it seems a bit contrary to my motivation to just cite it and proceed. Thus we get to the TL;DR.
TL;DR
Main Question: Is there a relatively short, fairly self contained proof that a noncompact manifold $M$ can be embedded in $\Bbb{R}^N$ for some $N$?
