PomeProposition 4.2.25.2733CF927914 Let $n\geq 2$ and $H=\{ z\in \mathbb C^ n:z_1=0\} $. Then the classical fine topology of $\mathbb C^ n=\mathbb R^{2n}$ induces the discrete topology on $H$. On the other hand, let $\tau $ be the coarsest topology on $\mathbb C^ n$ that contains the Euclidean topology and makes continuous every upper semicontinuous function $u\colon V\to [-\infty ,\infty )$, where $V\subset \mathbb C^ n$ is open, such that for every affine complex line $L$ the restriction of $u$ to each connected component of $L\cap V$ is subharmonic or identically $-\infty $. Then no point of $H$ is isolated in the topology induced on $H$ by $\tau $.
4.2.24 Complex lines and the choice of topology3AAB9C9B1384
On $\mathbb C^ n$, identified with $\mathbb R^{2n}$, the classical fine topology is generated by all subharmonic functions of $2n$ real variables. In 2 we shall instead use the upper semicontinuous functions whose restriction to each connected component of the intersection of their domain with an affine complex line is subharmonic or identically $-\infty $. On domains these are, apart from the constant $-\infty $, the plurisubharmonic functions of definition 5.8.1, and we shall see in example 5.47.17 that they generate the plurifine topology of definition 5.47.14 in the same way. In complex dimension one the line condition is planar subharmonicity, so the two topologies coincide; in higher dimensions they differ.
Proof. By example 3.4.15, the function $w\mapsto \log |w|$ is subharmonic on $\mathbb C$. Hence $z\mapsto \log |z_1|$, a function of the first two real coordinates, is subharmonic on $\mathbb R^{2n}$ by lemma 4.2.7; it is finite off $H$ and equals $-\infty $ on $H$. Thus $H$ is polar, and proposition 4.2.6 shows that the classical fine topology induces the discrete topology on $H$.
Now let $x\in H$, and let $N'$ be a neighborhood of $x$ for $\tau $. As in subsection 4.2.1, upper semicontinuity turns every condition of the form $u<b$ into a Euclidean open condition, and restrictions to smaller open sets preserve the complex-line condition; hence $N'$ contains a set
\[ N=V\cap \{ u_1>a_1\} \cap \dots \cap \{ u_ m>a_ m\} , \]
where $V$ is a Euclidean open neighborhood of $x$, the $u_ i$ are upper semicontinuous functions on $V$ satisfying the complex-line condition in the statement, and $u_ i(x)>a_ i$. Choose an affine complex line $\ell \subset H$ through $x$, which is possible since $n\geq 2$, and identify $\ell $ with $\mathbb C$. On the connected component of $\ell \cap V$ containing $x$, each restriction $u_ i|_\ell $ is subharmonic and is not identically $-\infty $, because $u_ i(x)$ is finite. Hence $\{ u_ i|_\ell >a_ i\} $ is open in the planar fine topology, and the intersection of these sets with that component is a planar fine neighborhood of $x$ contained in $N\cap \ell $. By proposition 4.2.5, applied in the plane, it contains points other than $x$. These points lie in $H\cap N'$, so $x$ is not isolated in $H$. $\square$
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