$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds.

The strong Whitney embedding theorem is usually stated as follows.

Theorem: If $M$ is a smooth $n$-dimensional manifold, then $M$ admits a smooth embedding into $\RR^{2n}$.

In fact, the theorem is stated in essentially this form in Whitney's original article "The self-intersections of a smooth $n$-manifold in $2n$-space". For definiteness, I will assume that all manifolds are Hausdorff, second countable, and smooth.

Question 1: Can we always take the embedding in the above theorem to be closed? If so, is there a reference for such a statement of the theorem?

It seems that Whitney's original proof produces an embedding whose image is not closed when $M$ is open. In fact, immediately after the construction, Whitney explicitly poses the following problem: "Does there exist an imbedding, for $M$ open, with no limit set?"

Having thought about the matter for a short while, I am inclined to believe that Whitney's trick (introduced in the aforementioned article by Whitney) allows the cancellation of infinitely many double points in a manner that preserves closed immersions. Is this correct? Or is my argument getting trapped in some pitfall?

My second question concerns possible dimensional restrictions in the above embedding theorem, stemming from the failure of Whitney's trick for $n=2$.

Question 2: Does every $2$-dimensional manifold embed in $\RR^4$? If so, can we also take the embedding to be closed in this case?

Here is the suggested proof in Whitney's article: "For $n=2$, we imbed the sphere, projective plane, or Klein bottle in $E^4$, and add the necessary number of handles to obtain the given manifold." I can see that this procedure should work for compact surfaces, but I am unable to carry it out in the non-compact case.

Finally, I would also be interested to hear about more recent, good references concerning Whitney's strong embedding theorem.