The Jordan-Brouwer separation theorem is a celebrated result of algebraic topology, which generalizes Jordan curve Theorem (it was proved independently by Lebesgue and Brouwer in 1911: see Dieudonné, A History of Algebraic Topology and Differential Topology for a detailed discussion of Lebesgue's and Brouwer's contributions).
The theorem has also a remarkable version for smooth (that is $C^{\infty}$) hypersurfaces in $\mathbb{R}^n$. This is the result as stated in Guillemin and Pollack, Differential Topology, Chapter 2, Sec.5:
The complement of the compact, connected, smooth hypersurface $X$ in $\mathbb{R}^n$ consists of two connected open sets, the "outside" $D_0$ and the "inside" $D_1$. Moreover the closure of $D_1$ is a smooth manifold $M$, with boundary $\delta M = X$.
My question is whether there is a $C^k$ version of this theorem (with $k \geq 1$), that is whether we can replace smooth by $C^k$ in the above statement. I do not know very much about differential topology, so I apologize if my question should be a trivial one.