The same topology presented on domains and then in charts on a complex manifold. Foundations additionally defines plurithinness; its full source statement is retained.
Definition 5.47.14.09A00D4B8096 Let $X$ be a complex manifold. The plurifine topology on $X$ is the coarsest topology on $X$ that contains the topology of $X$, which we call the Euclidean topology, and makes every $u\in \mathrm{PSH}(V)$, for every open set $V\subset X$, continuous as a map from $V$ with the induced topology to $[-\infty ,\infty )$. Sets that are open in this topology are called plurifine open, and neighborhoods in this topology are called plurifine neighborhoods. A set $E\subset X$ is plurithin at a point $a\in X$ if either $a$ does not belong to the Euclidean closure of $E\setminus \{ a\} $, or there is a plurisubharmonic function $\psi $ on a Euclidean open neighborhood of $a$ such that $\psi (a)$ is finite and
where the upper limit is taken with respect to the Euclidean topology.
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Definition 5.47.14.09A00D4B8096 The plurifine topology on $\Omega $ is the weakest topology making all $\mathbb {R}$-valued plurisubharmonic functions on $\Omega $ continuous.
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Definition 5.47.14.09A00D4B8096 The plurifine topology on $X$ is the topology with a base consisting of sets of the form $F^{-1}(V)$, where $U\subseteq X$ is an open subset, $F\colon U\rightarrow \Omega $ is a biholomorphic map, $\Omega \subseteq \mathbb {C}^ n$ is a domain for some $n\in \mathbb {N}$, and $V\subseteq \Omega $ is $\mathcal{F}$-open.
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