Let $M$ be a non-compact contractible bounded submanifold of the Euclidean space. Is it possible to find a compact submanifold $S$ of the Euclidean space such that $M$ is a submanifold of $S$?
As a positive example, let $M = \{(x,y) \in \mathbb{R}^2: (x,y) \in S^1, x>0, y>0\}$ be the portion of the 1-sphere $S^1$ lying in the positive orthant of $R^2$, and $S = S^1$.
Edit: the following example is wrong, as pointed out by a counter-example in a comment.
More generally, let $M$ be the image of an embedding $\gamma: (0,1) \to \mathbb{R}^2$ with $\gamma(0) \neq \gamma(1)$, assuming $M$ be bounded. Then, roughly speaking, $S$ can be obtained built joining the endpoints of $M$ by a "well-behaved" smooth curve $\gamma'$ that does not intersect $M$.
I posted a similar version of this question on mathoverflow, and I was redirected here.
I think the answer is "no" in general.
Suppose that $M$ is the interior of the Alexander horned sphere. How would you "complete" $M$ to a closed manifold $K$? This completion would need to contain the limit point of the 'arms', but that point would have no disk-like (or half-disk-like) neighborhood in $N$.