Further remarks on enlargements of generalized topologies (Q1928063)

Let \(\mu\) be a generalized topology on a set \(X\). Let \(\kappa\colon \mu \to \exp X\) (the power set of \(X\)) be an enlargement of \(\mu\). A subset \(A\) of \(X\) is said to be a \(\kappa_{\mu}\)-open set if \(x \in A \Rightarrow\) there exists a \(\mu\)-open set \(M\) such that \(x \in M\) and \(\kappa (\mu)\subset A\). \((X,\mu)\) is \(E\)-regular if for each \(x \in X\) and each \(\mu\)-open set \(U\) containing \(x\), there exists a \(\mu\)-open set \(V\) such that \(x\in V\), \(\kappa (V) \subset U\). Result: Let \((X,\mu)\) be an \(E\)-regular generalized topological space. Then every \(\mu\)-open set is a \(\kappa_{\mu}\)-open set. Let \(M \in \mu\). Then \(M\) is said to be \(\kappa\)-regular open if \(i_\mu (\kappa (M)) = M\). A generalized topology is a quasi topology on \(X\) if \(U, V \in \mu \Rightarrow(U\cap V) \in\mu\). Result: Let \(\mu\) be a quasi topology on \(X\). Let \(\kappa\) be an enlargement of \(\mu\). If \(K\) is monotonic, then the intersection of two \(\kappa\)-regular open sets is \(\kappa\)-regular open. Some similar results appear in this paper.