I have the impression that the answer to this should be elementary and maybe I am missing something obvious; however, a quick search in the literature gave no results.
Question. Does it exist an example of (possibly closed) non-orientable smooth manifold covered by exactly two charts? If so, what is a reference?
Of course, the intersection of these charts should be non-connected, because the transition function cannot change sign on a connected set.
Edit. Gae. S.'nice answer shows that the Klein bottle can be written as the union of two open cylinders (with disconnected intersection). So, let me ask the following variation of the above question:
Question 2. Does it exist a closed, non-orientable smooth manifold that can be written as the union of exactly two simply-connected charts?