I am trying to read Spivak's Differential Geometry and it seems that the first chapter contains many notions which are not treated rigorously.
Maybe I am not understanding enough, but the notion of identifying curves/points with each other isn't so clear to me.
He explained rigorously how to identify antipodal points with each other and that was a satisfactory explanation. However, it doesn't seem obvious to me how to rigorously define "identifications" in general.
For example, take the case of identifying the boundaries of two manifolds together. How would I make that rigorous?
Also, how would this work with other concepts of manifolds? It all seems very fuzzy to me. Intuitively, this would all go well, but rigorously, I just can't see anything.
Am I supposed to take it intuitively in this book, perhaps because it contains harder concepts beyond the scope of this book, or is there something important which I am missing?