\(f_1\)-sets and decomposition of \(R_IC\)-continuity (Q558273)

We know bitopological spaces where in a topological space \((X,\tau)\) there is considered a second topology. Recently several authors including the present ones studied ideal topological spaces. Here to \(\tau\) a set-ideal \(I\) is added: a family of subsets of \(X\) is called an ideal iff \(A\in I\) and \(B\subseteq A\) implies \(B\in I\), and \(A,B\in I\) implies \(A\cup B\in I\). Connecting \(\tau\) and \(I\) one defines a function \(A\to A^*(I):=\{x\in X| \forall U\in \underline{U}(x):U\cap A\not\in I\}\). Of course basic topological notions have been weakened, for instance \textit{N. Levine} [Amer. Math. Monthly 70, 36-41 (1963; Zbl 0113.16304)] called a subset \(A\) of a topological space \((X,\tau)\) semiopen iff \(A\subseteq Cl(int(A))\). In an ideal topological space \((X,\tau, I)\) other such notions are possible, for example: \(A\) is called \(\tau^*\)-closed iff \(A^*\subseteq A\), \(A\) is regular-\(I\)-closed iff \(A=(int(A))^*\). The authors define \(A\subseteq X\) to be an \(f_I\)-set iff \(A\subseteq (int(A))^*\), and they investigate the properties of \(f_I\)-sets. Now corresponding notions of continuity can be considered, namely by the authors: a function \(f:(X,\tau,I)\to (Y,\sigma)\) is said to be \(R_IC\)-continuous (resp. \(f_I\)-continuous, contra\(^*\)-continuous) iff for every \(V\in\sigma\), \(f^{-1}(V)\) is a regular-\(I\)-closed set (resp. \(f_I\)-set, \(\tau^*\)-closed set) of \((X,\tau,I)\). The main result of the paper shows that \(R_IC\)-continuity of \(f\) is equivalent to both, \(f_I\)-continuity and contra\(^*\)-continuity of \(f\).