Definition 4.2.2C067468FA3F0

Definition 4.2.2.C067468FA3F0 The classical fine topology on $\Omega $ is the coarsest topology on $\Omega $ that contains the Euclidean topology and makes every subharmonic function $u\in \mathrm{SH}(V)$, on every Euclidean open set $V\subset \Omega $, continuous as a map from $V$ with the induced topology to $[-\infty ,\infty )$. A set is finely open if it is open in this topology, and a fine neighborhood of a point is a neighborhood of it in this topology.