vector analysis: Stokes Thoerem says it extends to closed surfaces at $ mathbb {R} ^ 3 $?

Stokes Thoerem says $ oint _ { partial R} mathbf {F} cdot d mathbf {s} = iint_R nabla times mathbf {F} cdot d mathbf {S} $ for closed surfaces in $ mathbb {R} ^ 3 $?

My problem is that most of the theorem statements and Stoke's proofs seem to imply that the surface has a non-empty limit. Please note that I am do not talking about applying Stoke's theorem to closed surfaces by separating the surface, therefore showing that the surface integral of a curl is always $ 0 $.