Question How do I show that the Klein bottle is the connected sum of two $\Bbb{R}P^2$'s? More precisely; how do I construct an explicit continuous bijection between the Möbius strip and $\Bbb{R}P^2\setminus D^2$?
Thoughts If I can find an explicit continuous bijection between the Möbius strip $M$ and $\Bbb{R}P^2\setminus D^2$ where $D^2$ is the open disc, then I have a homeomorphism because $M$ is compact and $\Bbb{R}P^2\setminus D^2$ is Hausdorff. Then I will glue those $M$'s along their boundary.
I think of $M$ and $\Bbb{R}P^2\setminus D^2$ as the quotient spaces of $I\times I$, but I can't even see a continuous "deformation" between them visually so that I can cook up a continuous bijection according to the picture in my head.
Maybe the following picture of $\mathbb{R}P^2$ will help:
Just cut out the dark blue $D^2$ and a light blue moebius strip will remain.