I am trying to show that $\Bbb R^2$ induces a smooth structure on the Klein Bottle $K$ = $\Bbb R^2$/~, where the equivalence relation is: $\Bbb R^2$: ($x_1$, $x_2$) ∼ ($x_1 + n_1, 1 − x_2 + n_2$), for some ($n_1, n_2$) ∈ $\Bbb Z^2$.
(I believe this means that there exists a diffeomorphism from the Klein bottle to $\Bbb R^2$ modulo this particular equivalence relation).
The Goal (or my understanding of it anyway): Find charts that take an open sets in $\Bbb R^2$ to sets in the Klein Bottle's equivalence relation. This means, find $P_i: U_{open}$ $\subset$ $\Bbb R^2$ $\to$ $V \subset K$ = $\Bbb R^2$/~. Then, all we have to do is show that this charts follow the properties of 'smoothness':
$\quad$(i) each chart $P_i$ is injective,
$\quad$(ii) if ($V_\alpha \cap V_\beta) \neq \emptyset$ then $P^{-1}_\alpha(V_\alpha \cap V_\beta)$ and $P^{-1}_\beta(V_\alpha \cap V_\beta)$ are open in $\Bbb R^2$ and there exists a diffeomorphism from $V_\alpha \cap V_\beta$ to $P^{-1}_\beta(V_\alpha \cap V_\beta)$ given by $P^{-1}_\beta \circ P_\alpha$,
$\quad$(iii) $\cup_i P_i (U_i)= K$ which means the union of all the charts cover the whole manifold.
Finding the Charts: Using this equivalence relation, I would think the charts should look somewhat like so: $P_i: (x, y) \in \Bbb R^2$ $\to$ ($x_1 + n_1, 1 − x_2 + n_2$) such that $(n_1, n_2) \in \Bbb Z^2$.
My Question: I am still having trouble figuring out exactly what the charts should be and how many I will need to cover the whole manifold.
Any suggestions would be greatly appreciated.
Try using open balls in the plane of radius $1/2$ as your domain for the functions $P_i$. Since every point in the Klein bottle has a representative in the square $[0,1] \times [0,1]$, it will suffice to map the centers of these balls to $(0,0)$, $(1/2,0)$, $(0,1/2)$, and $(1/2,(1,2)$. (Try drawing these four open balls and then take into account that the parts of the ball which land outside this square still cover points of the Klein bottle. One useful way to think about this is the model where opposite sides are identified, although the top and bottom sides are identified with a flip of the edge.) Once you are convinced these balls are mapped by homeomorphisms, you need to study what happens to points which lie in two balls. This is where some work will need to be done.
There are other ways to approach this problem, of course. The one which is close to this is when one has a nice (discrete) group action on a space you already know has a smooth structure and the group acts by diffeomorphisms. In this context, the Klein bottle admits a smooth structure because there is a group acting by (sometimes orientation reversing) affine maps on the plane and the Klein bottle is the resulting quotient object (called the orbit space).