their endpoints such that $\widehat{ M} \cup
\gamma_1 \cup \cdots \cup \gamma_r$ is connected. Since $\Omega
\backslash K$ is connected, we can insure that each $\gamma_k$ is contained in $\Omega \backslash K$.

connecting-tubes.pstex\_t

We can then modify $\widehat{ M}$ in the following way: we cut out a very small ball in $\widehat{ M}$ around each endpoint of every $\gamma_k$, and we link up the connected components of the excised hypersurface with $r$ thin tubes $\simeq {\mathbb{R}}\times S^{ 2n - 2}$ almost parallel to the $\gamma_k$, smoothing out the corners appearing near the endpoints. The resulting hypersurface $M$ is $\mathcal{ C}^\infty$ and connected. Since each $\gamma_k$ is either contained in $\widehat{ \Omega} \cup \widehat{ M}$ or in ${\mathbb{R}}^{
2n} \big \backslash \widehat{ \Omega }$, a new open set $D$ with $\partial D = M$ is obtained by either deleting from $\widehat{
\Omega}$ or adding to $\widehat{ \Omega}$ the thin tube around each $\gamma_k$. All the tubes around the $\gamma_k$ which are contained in ${\mathbb{R}}^{ 2n} \big \backslash \widehat{ \Omega}$ constitute thin open tunnels between the components of $\widehat{ \Omega}$, whence $D$ is connected.

On a general complex space $X$, the idea is to embed a neighborhood of $\overline{ \Omega}$ smoothly into 