By Whitney's embedding theorem, every 2-dimensional manifold (aka. a surface) can be embedded into $\Bbb R^4$.
Now, the Wikipedia page on that theorem states in this paragraph that we can even embedd into $\Bbb R^3$ if the surface is
- compact and orientable, or
- compact and with non-empty boundary.
This seems to suggest that there is an orientable but non-compact surface that does not embedd into $\Bbb R^3$. Is this true?