\(g\)-closed sets in ideal topological spaces (Q949852)

Given a topological space \((X,\tau)\), an ideal \(\mathcal I\) in the powerset \(\mathcal P(X)\), and a subset \(A\subseteq X\), let \(A^*=\{x\in X:U\cap A\notin \mathcal I\;\text{for each open neighborhood }U\) of