In learning differential topology I've been exposed with two methods of defining and working with manifolds: The more concrete but initially less general approach of Guillemin and Pollack, where all manifolds sit inside an ambient Euclidean space; and the more abstract approach of most other books, like Lee's ISM, where manifolds are defined intrinsically via charts and atlases. One benefit of the second approach is that one can define a manifold by gluing (e.g. the Mobius strip or Klein bottle) without worrying if the prescribed gluing is possible in some Euclidean space.
But certain exercises in Guillemin and Pollack do seem to define manifolds by gluing (Exercise 1.6 in Chapter 2 does this with the Mobius strip). I don't see how this procedure can be rigorous unless the more abstract chart-atlas definition is used, so I wanted to know how Guillemin and Pollack expect their readers to formalize this construction if they have not been exposed to charts and atlases. For example, the reader is at one point asked to build the Mobius strip as a manifold with boundary by gluing opposite ends of a square together. Is it possible to do this rigorously when we're working with the definition that all manifolds must sit inside Euclidean space?