Prove that the real projective line cannot be embedded into Euclidean space Can the real projective line $RP^1$ be embedded into $\mathbb{R}^n$ for any $n$? 
At first, I thought it could because $RP^2$ can be embedded in four-dimensional space, and $RP^1$ seemed like a simpler object to deal with than $RP^2$. But after drawing diagrams and attempting to find an embedding, I'm getting less and less sure that it can be embedded... unrigorously it always seems to require that 'point at infinity' that comes from say the one-point compactification of $\mathbb{R}$, so is there a way that I could prove this one way or the other?

In lieu of a thousand words, here's an embedding of the real projective line (the set of lines in the plane through the origin) in the Cartesian plane:

