compact and $S$ [*noncompact*]{}, take a point $s \in S$ and set $K := R \times \{ s\}$; by [@fo1981], there exists a meromorphic function function having a pole of order $1$ at $s$, whence $\mathcal{ O} ( X)$ does not extend through $K$. Consequently, in the category of strong Levi-form assumptions, $(n-1)$-convexity is sharp.

For the theorem, the main strategy of proof consists of performing holomorphic or meromorphic extension entirely within the regular part of $X$.

With $X$, $\Omega$ and $K$ as in Theorem 2.2, holomorphic or meromorphic functions on $\big[ \Omega \backslash K
\big]_{ \rm reg}$ extend holomorphically or meromorphically to $\Omega_{\rm reg}$.

Notice that both $\big[ \Omega \backslash K \big]_{\rm reg}$ and $\Omega_{\rm reg}$ are connected (footnote 3). Then by , extension immediately holds to $\Omega$. This yields Theorem 2.2 if one takes the proposition for granted; Sections 3 and 4 below are devoted to prove this proposition.

For meromorphic extension, one could in principle well avoid the assumption of normality. In the case of meromorphic extension, we get a general result valid for reduced spaces without further local assumptions.

Let $X$ be a globally irreducible $(n-1)$-complete reduced complex space of pure dimension $n\geqslant 2$. Then for 