Example 5.47.177E1189000F74

Example 5.47.17.7E1189000F74 Let $n\ge 2$ and $H=\{ z\in \mathbb C^ n:z_1=0\} $. Then $H$ is pluripolar, being the pole set of $\log |z_1|^2$ (proposition 5.10.3), but no point of $H$ is isolated in the topology induced on $H$ by the plurifine topology. This is the second assertion of proposition 4.2.25, because the topology $\tau $ of that proposition is the plurifine topology. Indeed, every $u\in \mathrm{PSH}(V)$, with $V\subset \mathbb C^ n$ open, satisfies the conditions in the definition of $\tau $, so every plurifine open set is open for $\tau $. Conversely, let $u\colon V\to [-\infty ,\infty )$ be as in the definition of $\tau $. By definition 5.8.1, $u$ is plurisubharmonic on the union $V'$ of the connected components of $V$ on which it is not identically $-\infty $. Hence its superlevel sets $\{ u>t\} =\{ z\in V':u(z)>t\} $, $t\in \mathbb R$, are plurifine open, while its sublevel sets are Euclidean open. Thus $u$ is continuous for the plurifine topology, and every $\tau $-open set is plurifine open.

Consequently, neither conclusion of proposition 4.2.6 carries over to pluripolar sets: $H$ is not plurifinely discrete, and for $a\in H$ the pluripolar set $H\setminus \{ a\} $ is not plurifine closed, since every plurifine neighborhood of $a$ meets it.