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 $$0$$.