in $\big( \Omega \backslash
K\big) \cap X_{\rm reg}$.

Let $X$, $\Omega$ and $K$ be as in Theorem 2.2. Then there is a domain $D \Subset \Omega$ containing $K$ such that $M := \partial
D \cap X_{\rm reg}$ is a $\mathcal{ C}^\infty$ [connected]{} hypersurface of $X_{\rm reg}$.

Suppose first that $X = {\mathbb{C}}^n$. Let ${\sf d}$ be a regularized distance function ([@st1970]) for $K$, [*i.e.*]{} a $\mathcal{
C}^\infty$ real-valued function with $K = \{ {\sf d} = 0 \}$ and $\frac{ 1}{ c}\, {\rm dist} \, ( x, K) \leqslant {\sf d} ( x)
\leqslant c \, {\rm dist}\, ( x, K)$ for some constant $c >1$, where ${\rm dist}$ is the Euclidean distance in ${\mathbb{R}}^{ 2n}$. By Sard’s theorem, there are arbitrarily small $\varepsilon >0$ such that $\widehat{ M} := \{ {\sf d} = \varepsilon \}$ is a $\mathcal{
C}^\infty$ hypersurface of ${\mathbb{R}}^{ 2n}$ bounding the open set $\widehat{
\Omega} := \{ {\sf d} < \varepsilon \}$ which satisfies $K \subset
\widehat{ \Omega} \Subset \Omega$. However, since $\widehat{ M}$ need not be connected, we must modify it.

To this aim, we pick finitely many disjoint closed simple $\mathcal{
C}^\infty$ arcs $\gamma_1, \dots, \gamma_r$ which meet $\widehat{ M}$ transversally only at 