First a small disclaimer that I have been introduced to manifolds but I am not extremely comfortable with them in the general case yet, however I am taking a course on curves and surfaces (which is almost over) and thus am quite familiar with them (i.e. I know what manifolds are, but I am more used to working with curves and surfaces in $\mathbb R^3$ and $\mathbb R^2$).
My question is why is must we restrict ourselves to defining manifolds as homeomorphic to open subsets of $\mathbb R^n$. Allowing us to use closed sets would allow us, for example to cover $S^2$ with a single surface patch, as opposed to two, which seems like an attractive property.
I'm going to go on to study more advanced differential geometry very soon and I would love to have a good motivation for this property. Thanks in advance!