Regular sets in generalized topological spaces (Q1928068)

For a set \(X\) with \(\mathcal P(X)\) as the power set, a subcollection \(\mu \subseteq \mathcal P(X)\) is said to be a generalized topology if \(\emptyset \in \tau\) and any union of elements of \(\tau\) belongs to \(\tau\); \((X, \mu )\) is called a generalized topological space (GTS-in short). If \(X \in \mu\), \((X, \mu )\) is called a strong GTS; if \(\mu\) is closed under finite intersections, \((X, \mu )\) is called a quasi-topological space; the concepts of \(c_\mu (A)\) and \(i_\mu (A)\), for \(A \subseteq X\) come very naturally; \(A \subseteq X\) is said to be \(\alpha\)-regular if \(A = i_\mu c_\mu i_\mu (A)\). \(\sigma\)-regular, \(\pi\)-regular, \(\beta\)-regular sets are also defined by using \(i_\mu\) and \(c_\mu\). A theorem characterizing \(\sigma\)-regular sets and \(\sigma\)-clopen sets of a GTS \((X, \mu )\) is stated and proved; similar necessary and sufficient conditions for \(\beta\)-regular sets and \(\beta\)-clopen sets, \(\pi\)-regular sets in a GTS and \(\pi\)-clopen sets in extremally \(\pi\)-disconnected spaces are enunciated and proved. \(\mu\)-locally \(\mu\)-regular sets are also defined and a characterization theorem for such sets is proved. A characterization theorem for \(\mu\)-locally \(\sigma\)-regular sets in a quasitopological space is also provided; another such, for \(\mu\)-locally \(\pi\)-regular sets is also established. The last theorem gives a characterization of \(\mu\)-open sets in terms of \(\mu\)-locally, \(\mu\)-closed, \(\mu\)-locally \(\sigma\)-closed and \(\mu\)-locally \(\sigma\)-regular sets in a quasi topological space.