Open subset of Euclidean space can't have non-spherical manifold compactification. The title is not complete, since it would be too long. Consider the following statement:

Let $U \subset \mathbb{R}^n$ be open, connected and such that its one-point compactification is a manifold. Then, this compactification must be (homeomorphic to) the sphere $S^n$.

Is the statement above true? If so, why?

I can imagine an ellementary approach only for  the special case of $\mathbb{R^2}  $ and $\mathbb{R}$.
For $\mathbb{R^2}$:We know that all the compact surfaces arise from adding to the sphere a finite amount of handles or Mobius-strips. In any case, if you remove a point from a compact surface which is something of the above but not a sphere then you wouldn't get something homemoprhic to $\mathbb{R^2}$. So the only compactification could be to a sphere.
Similar arguments goes for $\mathbb{R}$ since the only compact $1$ manifolds are a closed line segment and the circle. 
