This question arose during my Differential Geometry course. Possibly there is an obvious answer, but I do not see it, and I could not find it in the literature. The same question was asked yesterday on MSE, but it did not get much attention.
Question. Does it exist a closed, non-orientable smooth manifold that can be written as the union of exactly two simply-connected charts? If so, what is a reference?
Of course, the intersection of the two charts must be disconnected. As noted in the linked MSE question, the open Möbius band and the closed Möbius band give examples in the open case and in the non-empty boundary case, respectively. Moreover, the Klein bottle is covered by two charts, both homeomorphic to cylinders, but they are not simply-connected.