PomeCorollary 5.15.15.C94ADCA6E19D Let $\Omega \subseteq \mathbb {C}^ n$ be an $\mathcal{F}$-open subset and $x\in \Omega $. Then $x$ has a compact $\mathcal{F}$-neighborhood contained in $\Omega $.
Corollary 5.15.15C94ADCA6E19D
Proof. By theorem 5.15.11, it suffices to prove the assertion after replacing $\Omega $ by a smaller $\mathcal F$-open neighborhood of $x$. Thus we may assume that
\[ \Omega =\{ y\in U:\varphi (y)>0\} \]
for some open set $U\subseteq \mathbb C^ n$ and some $\varphi \in \mathrm{PSH}(U)$.
Take a compact neighborhood $K$ of $x$ in $U$. Now $\{ y\in K: \varphi (y)\geq \varphi (x)/2\} $ is a compact $\mathcal{F}$-neighborhood of $x$ contained in $\Omega $. $\square$
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