This questions arises from working through the 2015 Heraeus lectures on gravity, specifically tutorial 2 exercise 2, so this is why I post it as a physics question although it is purely mathematical.
When we represent the Moebius strip as a rectangle, we draw opposite sense arrows on two opposite sides that describe how to glue the sides, i.e. Moebius strip as a rectangle. Now if I want to show it is a 2d topological manifold, I need to find a set of charts that cover it, so basically some open set(s) in the "Moebius rectangle" together with the corresponding homeomorphism(s) from each to $\mathbb{R}^2$. Here are my questions:
- Why no single chart can do this;
- why (and what) two charts would suffice;
I can see there is a problem with the twist on the top and bottom sides, like when I try to draw a ball there it gets ripped in two, but that doesn't help me answer the above.
The fact that two suffice is easy, just find two. (Look at a picture of it.)
To show that one chart does not suffice, note that if one chart could cover it, then it could be oriented by the orientation inherited from $\Bbb R^2$. However, it is known that it is not orientable.