How can one show that the Klein bottle can be immersed in $\mathbb{R}^3$ and embedded in $\mathbb{R}^4$?
We define the Klein bottle as the quotient space of $I^2=[0,1]\times [0,1]$ under the relation $\sim$ for which $(0,y)\sim (1,1-y)$ and $(x,0)\sim (x,1)$.
If we found a continuous $f:I^2\to R^k$ which is constant over the fibers of $\pi:I^2\to I^2/{\sim}$ then we'd have a continuous $g:I^2/{\sim}\to\mathbb{R}^k$ such that $g\pi=f$.
What do I need for $g$ to be an immersion? And to be an embedding?
How can I show that there is an immersion for $k=3$ and an embedding for $k=4$ (without finding it explicitly)?