every domain $\Omega
\subset X$ and every compact set $K \subset \Omega$ with $[\Omega
\backslash K]_{\rm reg}$ connected, meromorphic functions on $\Omega
\backslash K$ extend meromorphically and uniquely to $\Omega$[:]{} $$\mathcal{M}_X(\Omega\backslash K)
=
\mathcal{M}_X(\Omega)\big\vert_{\Omega\backslash K}.$$ If moreover the data lie in $\mathcal{O}_X(\Omega\backslash 
K)$, the extension is weakly holomorphic.

The proof, also relying upon an application of Proposition 2.3, is postponed to Section 5; an example in §5.1 shows that requiring only that $\Omega \backslash K$ is connected does not suffices.

For the proposition, the main difficulty is that $X_{\rm sing}$ can in general cross $\Omega \backslash K$. We will approach $X_{\rm sing}$ from the regular part and fill in progressively $\Omega_{\rm reg}$ by means of the super-level sets of a suitable modification $\mu$ of the exhaustion $\rho$, such that $\mu$ is still strongly $(n-1)$-convex but exhausts only $X_{\rm reg}$ in a neighborhood of $\overline{
\Omega}$. To verify that the extension procedure devised in [@mp2006b] can be performed, preliminaries are required.

§3. Geometrical preparations {#geometrical-preparations .unnumbered}
============================

3.1. Smoothing out the boundary {#smoothing-out-the-boundary .unnumbered}
-------------------------------

To launch the filling procedure, we want to view the connected open set $\Omega \backslash K$ as a neighborhood of some convenient connected hypersurface $M$ contained 