Why is it that $\partial S^2=\emptyset$

Question

While doing some differential geometry I came across the following application of the Stokes Theorem. $$\int_{S^2}d(xdy)=\int_{\partial S^2}xdy$$ Now, the professor baldly affirms that $\partial S^2=\emptyset$. I don't understand why is that. Isn't it that $$\partial S^2=\{(x,y,z)\in\mathbb{R}^3|x^2+y^2+z^2=1\}$$