having finitely many connected components which are contained in $\Psi \big( [ \Omega \backslash K ]_{\rm reg} \big)$. Importantly, we can construct the thin connecting tubes so that they [*lie all entirely inside*]{} $\Psi \big( \big[ \Omega \backslash K \big]_{\rm
reg} \big)$, thanks to the fact that $\Psi \big( \Omega_{\rm reg}
\big)$ is locally (arcwise) connected, also near points of $\Psi \big(
\Omega_{\rm sing} \big)$. Then the remaining arguments are the same and we put everything back to $X$ [*via*]{} $\Psi^{ - 1}$, getting a connected $\mathcal{ C}^\infty$ hypersurface $M \subset [\Omega
\backslash K]_{\rm reg}$ and a domain $D$ with $K \subset D \Subset
\Omega$. (We remark that normality of $X$ was crucially used.)

As we said, we will perform the filling procedure entirely inside $X_{\rm reg}$. This is possible thanks to an idea of Demailly which consists of modifying the initial exhaustion $\rho$ so that $X_{\rm
sing}$ is put at $- \infty$. A recent application of this idea also appears in [@df2007].

3.3. Putting $X_{\rm sing}$ into a well {#putting-x_rm-sing-into-a-well .unnumbered}
---------------------------------------

By Lemma 5 in [@de1990], there exists an [*almost plurisubharmonic function*]{}[^4] $v$ on $X$ which is $\mathcal{ C}^\infty$ on $X_{\rm
reg}$ and has poles along $X_{\rm sing}$: 