Could someone please explain what "embedding" means (maybe a more intuitive definition)? I read that the Klein bottle and real projective plane cannot be embedded in ${\mathbb R}^3$, but is embedded in ${\mathbb R}^4$. Aren't those two things 3D objects? If so, why aren't they embedded in ${\mathbb R}^3$? Also, I have come across the word "immersion". What is the difference between "immersion" and "embedding"?
Thanks.
Basically an abstract surface has, at every point two independent directions along the surface. Or even better, there is an entire circle's worth of rays coming out of each point.
An immersion is, roughly, a map of the surface into a bigger manifold (such as $\mathbb R^n$) where there are still two dimensions worth of rays emanating out of each point. So for the usual immersion of a Klein bottle into $\mathbb R^3$, at the circle of self-intersection, each sheet still retains its two dimensional character. So it is an immersion. If you were to instead map the Klein bottle into $\mathbb R^3$ by mapping everything to a point, that would not be an immersion.