I am going over these notes on differential forms and I learn a 1-form is defined the following way:
A differential 1-form (or simply a differential or a 1-form) on an open subset of $\mathbb R^2$ is an expression $F(x, y)dx+G(x, y)dy$ where F, G are $\mathbb R$-valued functions on the open set.
My question is what is the significance of open set in the definition?
I don't know the actual reason why your definition focuses on open subsets. I can't read the mind of its author. But I have a few ideas that I think are plausible.
Manifolds have differential forms on them, but you usually do calculations with these forms on charts, i.e. in $\Bbb R^n$ or open subsets thereof.
Differential forms are tightly linked with tangent vectors, and derivatives. No one wants to bother with what tangent vectors and derivatives, and by extension differential forms, do on a boundary unless they absolutely have to. So we just ignore boundaries all together and concentrate on interiors.