PomeTheorem 5.15.10 (H. Cartan).458C93F5F6CD Consider a set $E\subseteq \Omega $ and $x\in E$. Then the following are equivalent:
15B95CB674C7$E$ is an $\mathcal{F}$-neighborhood of $x$;
53048930270F$\Omega \setminus E$ is thin at $x$.
Theorem 5.15.10 (H.Cartan)458C93F5F6CD
Proof. 2 $\implies $ 1. Assume 2. We may assume that $x\in \overline{\Omega \setminus E}$. Otherwise, our assertion follows from proposition 5.15.2.
By proposition 5.15.6, there is an open neighborhood $U$ of $x$ in $\Omega $ and $\varphi \in \mathrm{PSH}(\mathbb {C}^ n)$ such that
\[ \varphi (x)>\sup _{y\in U\cap (\Omega \setminus E)} \varphi (y)\eqqcolon \lambda . \]
Let $F=\{ y\in \Omega : \varphi (y)>\lambda \} $. Then $x\in F$ and $F$ is $\mathcal{F}$-open by example 5.15.3. Moreover, $U\cap F\subseteq E$. By proposition 5.15.2, we conclude 1.
1 $\implies $ 2. Assume 1. We may always replace $E$ by smaller $\mathcal{F}$-neighborhoods of $x$. In particular, we may assume that $E$ has the following form
\[ \{ y\in U:\varphi _1(y)>\lambda _1,\ldots ,\varphi _ m(y)>\lambda _ m\} , \]
where $U\subseteq \Omega $ is an open neighborhood of $x$, and $\varphi _1,\ldots ,\varphi _ m$ are $\mathbb {R}$-valued psh functions on $\Omega $, and $\lambda _1,\ldots ,\lambda _ m\in \mathbb {R}$. Since $x\in U$, the set $\Omega \setminus U$ is thin at $x$. Moreover,
\[ \Omega \setminus E=(\Omega \setminus U)\cup \bigcup _{i=1}^ m\{ y\in U:\varphi _ i(y)\leq \lambda _ i\} . \]
By 5.15.9, it is enough to prove that each set $\{ y\in U:\varphi _ i(y)\leq \lambda _ i\} $ is thin at $x$. Thus we may assume $m=1$. In this case, $\Omega \setminus E$ is clearly thin at $x$. $\square$
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