Approximation of a Reifenberg-flat set by a smooth surface

We show that if $E\text{i}{\mathbb{R}}^n$ is a Reifenberg flat set $E$ of dimension $d$ at scale $r_0$, we can find a smooth surface ${\mathrm{\Sigma }}_0$ of dimension $d$ which is close to $E$ at the scale $r_0$. When $E$ is a Reifenberg flat set, this allows us to apply a result of G. David and T. Toro [Memoirs of the AMS 215 (2012), 1012], and get a bi-H"older homeomorphism of ${\mathbb{R}}^n$ that sends ${\mathrm{\Sigma }}_0$ to $E$. If in addition $d=n-1$ and $E$ is compact and connected, then ${\mathrm{\Sigma }}_0$ is orientable, and ${\mathbb{R}}^n\text{sm}E$ has exactly two connected components, which we can approximate from the inside by smooth domains.